This is a text file containing the computation of the eta invariants of manifolds in the Snappea census with Chern-Simons invariant zero. If the eta invariant is integral, then the manifold might bound geometrically. There are 41 such manifolds on this list. None are known to bound geometrically. This file is easily searched with any text editor and the nonintegral eta manifolds are flagged. 1. m007(3,1): pr inv Volume: 1.014941606409653625021202554 Chern-Simons: 0.E-58 1. m007(3,1): pr eta inv Eta invariant: -2.074338426365847919000465204 E-59 3*eta - 2*CS: -1.000000000000000000000000000 1. m007(3,1): compute invariant_trace_field Invariant trace field: x^2 - x + 1 [0, 1] -3 R(1) = 0.500000 + 0.866025*I 1. m007(3,1): pr arithmetic Invariant trace field: x^2 - x + 1 [0, 1] -3 R(1) = 0.500000 + 0.866025*I Invariant quaternion algebra: Hilbert Symbol: (4*x - 8, 12*x - 12) Real ramification: [] Finite ramification: [2, [2, 0]~, 1, 2, [1, 0]~] [3, [1, 1]~, 2, 1, [1, 1]~] Integer traces: YES Arithmetic: YES Borel Regulator: not computed (shape field required) 1. m036(-3,2): pr inv Volume: 2.029883212819307250042405108 Chern-Simons: 0.E-58 1. m036(-3,2): pr eta inv Eta invariant: -2.765784569998955286793465568 E-60 3*eta - 2*CS: 0.E-78 1. m036(-3,2): com inv Invariant trace field: x^2 - x + 1 [0, 1] -3 R(1) = 0.500000 + 0.866025*I 1. m036(-3,2): pr arith Invariant trace field: x^2 - x + 1 [0, 1] -3 R(1) = 0.500000 + 0.866025*I Invariant quaternion algebra: Hilbert Symbol: (5*x - 1, -6*x + 6) Real ramification: [] Finite ramification: [2, [2, 0]~, 1, 2, [1, 0]~] [3, [1, 1]~, 2, 1, [1, 1]~] Integer traces: YES Arithmetic: YES Borel Regulator: not computed (shape field required) 1. m034(-4,1): pr inv Volume: 2.195964118694024514602807723 Chern-Simons: -1.062061274088301517939079976 E-58 1. m034(-4,1): pr eta inv Eta invariant: -6.084726049464227446306879842 E-60 3*eta - 2*CS: 0.E-78 1. m034(-4,1): com inv Invariant trace field: x^8 - x^7 - x^6 + 3*x^5 - 3*x^4 + 3*x^3 - x^2 - x + 1 [2,3] -14174503 R(3) = -0.0744187 + 0.997227*I 1. m034(-4,1): pr arith Invariant trace field: x^8 - x^7 - x^6 + 3*x^5 - 3*x^4 + 3*x^3 - x^2 - x + 1 [2,3] -14174503 R(3) = -0.0744187 + 0.997227*I Invariant quaternion algebra: Hilbert Symbol: (-x^7 + 3*x^6 + 3*x^5 - 5*x^4 + 4*x^3 - 2*x^2 + 2*x + 1, -x^7 + 4*x^6 - x^5 - 9*x^4 + 13*x^3 - 5*x^2 - 2*x + 3) Real ramification: [1, 2] Finite ramification: [2, [3, 1, 1, 0, 0, 0, 0, 0]~, 1, 2, [1, 0, 0, 1, 0, 0, 1, 0]~] [7, [1, -3, 1, 0, 0, 0, 0, 0]~, 1, 2, [1, 2, -3, -1, -3, 2, 1, 0]~] Integer traces: YES Arithmetic: NO 1. m160(-3,2): pr inv Volume: 2.595387593686742138301993834 Chern-Simons: 5.310306368395686149000000000 E-59 1. m160(-3,2): pr eta inv Eta invariant: -0.6666666666666666666666666666 NONINTEGRAL ETA 3*eta - 2*CS: -2.000000000000000000000000000 1. m160(-3,2): com inv Invariant trace field: x^4 - x^3 - x^2 - x + 1 [2, 1] -507 R(3) = -0.651388 + 0.758745*I 1. m160(-3,2): pr arith Invariant trace field: x^4 - x^3 - x^2 - x + 1 [2, 1] -507 R(3) = -0.651388 + 0.758745*I Invariant quaternion algebra: Hilbert Symbol: (x^3 - 3*x^2 + x - 1, 2*x^3 - 3*x^2 - 3*x + 2) Real ramification: [1, 2] Finite ramification: none Integer traces: YES Arithmetic: YES Borel Regulator: not computed (shape field required) 1. m082(1,3): pr inv Volume: 2.786804556415568521855509429 Chern-Simons: 5.310306368395686149000000000 E-59 1. m082(1,3): pr eta inv Eta invariant: 4.203992548211801709521356906 E-59 3*eta - 2*CS: 1.000000000000000000000000000 1. m082(1,3): com inv Invariant trace field: x^4 - x^3 - 2*x^2 - x + 1 [2, 1] -1156 R(3) = -0.780776 + 0.624810*I 1. m082(1,3): pr arith Invariant trace field: x^4 - x^3 - 2*x^2 - x + 1 [2, 1] -1156 R(3) = -0.780776 + 0.624810*I Invariant quaternion algebra: Hilbert Symbol: (-1/4*x^3 - 3/4*x^2 + 7/4*x - 2, 6*x^3 - 21/2*x^2 - 15/2*x + 15/2) Real ramification: [] Finite ramification: [2, [1, 1, 1, 0]~, 1, 2, [1, 0, 1, 0]~] [3, [3, 0, 0, 0]~, 1, 4, [1, 0, 0, 0]~] Integer traces: NO Arithmetic: NO Borel Regulator: not computed (shape field required) 1. m078(5,1): pr inv Volume: 2.816179876317820792596408257 Chern-Simons: 1.062061273492872714000000000 E-58 1. m078(5,1): pr eta inv Eta invariant: -7.746962576278819314051088396 E-59 3*eta - 2*CS: -1.000000000000000000000000000 1. m078(5,1): com inv Invariant trace field: x^6 - x^4 - x^3 - x^2 + 1 [2, 2] 52441 R(4) = -0.127050 + 0.991896*I 1. m078(5,1): pr arith Invariant trace field: x^6 - x^4 - x^3 - x^2 + 1 [2, 2] 52441 R(4) = -0.127050 + 0.991896*I Invariant quaternion algebra: Hilbert Symbol: (x^2 - x - 3, -6*x^5 + 12*x^3 + 4*x^2 - 6*x - 2) Real ramification: [1, 2] Finite ramification: [2, [1, 1, 1, 0, 0, 0]~, 1, 2, [1, 1, 1, 1, 1, 0]~] [2, [1, 1, 1, 1, 1, 0]~, 1, 4, [1, 1, 1, 0, 0, 0]~] Integer traces: YES Arithmetic: NO Borel Regulator: not computed (shape field required) 1. m168(2,3): pr inv Volume: 2.846510545640650382556241638 Chern-Simons: 0.E-58 1. m168(2,3): pr eta inv Eta invariant: -4.646518072899917091105094031 E-59 3*eta - 2*CS: -1.000000000000000000000000000 1. m168(2,3): com inv Invariant trace field: x^6 - x^5 - x^4 + 2*x^3 - x + 1 [0, 3] -10571 R(3) = 1.07394 + 0.558752*I 1. m168(2,3): pr arith Invariant trace field: x^6 - x^5 - x^4 + 2*x^3 - x + 1 [0, 3] -10571 R(3) = 1.07394 + 0.558752*I Invariant quaternion algebra: Hilbert Symbol: (x^5 - x^4 + 2*x^3 + x^2 - 3*x - 1, 6*x^5 - 6*x^3 - 2) Real ramification: [] Warning: ramification test aborted Finite ramification: [-2, [2, 0, 0, 0, 0, 0]~, 1, 6, [1, 0, 0, 0, 0, 0]~] [13, [-2, -4, 1, 0, 0, 0]~, 1, 2, [6, -5, 0, 3, 1, 0]~] Integer traces: YES Arithmetic: NO Borel Regulator: not computed (shape field required) 1. m100(2,3): pr inv Volume: 2.882494387251527860472482762 Chern-Simons: 1.593091911132452276962830073 E-58 1. m100(2,3): pr eta inv Eta invariant: 2.655153185220753794901910049 E-59 3*eta - 2*CS: 0.E-77 1. m100(2,3): com inv Invariant trace field: x^6 - x^4 - x^3 - x^2 + 1 [2, 2] 52441 R(3) = -0.930403 + 0.366538*I 1. m100(2,3): pr arith Invariant trace field: x^6 - x^4 - x^3 - x^2 + 1 [2, 2] 52441 R(3) = -0.930403 + 0.366538*I Invariant quaternion algebra: Hilbert Symbol: (x^4 - 4*x^2, 6*x^5 - 6*x^4 - 2*x^3 - 2*x^2 - 4*x + 6) Real ramification: [] Finite ramification: [2, [1, 1, 1, 0, 0, 0]~, 1, 2, [1, 1, 1, 1, 1, 0]~] [2, [1, 1, 1, 1, 1, 0]~, 1, 4, [1, 1, 1, 0, 0, 0]~] Integer traces: YES Arithmetic: NO Borel Regulator: not computed (shape field required) 1. m149(-4,1): pr inv Volume: 3.044824819228960875063607662 Chern-Simons: 6.456558322533965110000000000 E-59 1. m149(-4,1): pr eta inv Eta invariant: -0.3333333333333333333333333333 NONINTEGRAL ETA 3*eta - 2*CS: 0.E-67 1. m149(-4,1): com inv Invariant trace field: x^2 - x + 1 [0, 1] -3 R(1) = 0.500000 + 0.866025*I 1. m149(-4,1): pr arith Invariant trace field: x^2 - x + 1 [0, 1] -3 R(1) = 0.500000 + 0.866025*I Invariant quaternion algebra: Hilbert Symbol: (x - 5, 30*x - 24) Real ramification: [] Finite ramification: [2, [2, 0]~, 1, 2, [1, 0]~] [7, [2, 1]~, 1, 1, [-3, 1]~] Integer traces: YES Arithmetic: YES Borel Regulator: not computed (shape field required) 1. m188(2,3): pr inv Volume: 3.044824819228960875063607662 Chern-Simons: 1.858607229654527656474705121 E-58 1. m188(2,3): pr eta inv Eta invariant: 4.425255311174060425354059855 E-59 3*eta - 2*CS: 0.E-77 1. m188(2,3): com inv Invariant trace field: x^2 - x + 1 [0, 1] -3 R(1) = 0.500000 + 0.866025*I 1. m188(2,3): pr arith Invariant trace field: x^2 - x + 1 [0, 1] -3 R(1) = 0.500000 + 0.866025*I Invariant quaternion algebra: Hilbert Symbol: (3*x - 4, -10) Real ramification: [] Finite ramification: [2, [2, 0]~, 1, 2, [1, 0]~] [5, [5, 0]~, 1, 2, [1, 0]~] Integer traces: YES Arithmetic: YES Borel Regulator: not computed (shape field required) 1. m247(-1,3): pr inv Volume: 3.044824819228960875063607662 Chern-Simons: -1.327576592098921537000000000 E-59 1. m247(-1,3): pr eta inv Eta invariant: 0.6666666666666666666666666666 NONINTEGRAL ETA 3*eta - 2*CS: 1.000000000000000000000000000 1. m247(-1,3): com inv Invariant trace field: x^2 - x + 1 [0, 1] -3 R(1) = 0.500000 + 0.866025*I 1. m247(-1,3): pr arith Invariant trace field: x^2 - x + 1 [0, 1] -3 R(1) = 0.500000 + 0.866025*I Invariant quaternion algebra: Hilbert Symbol: (4*x - 8, 24*x - 12) Real ramification: [] Finite ramification: [2, [2, 0]~, 1, 2, [1, 0]~] [3, [1, 1]~, 2, 1, [1, 1]~] Integer traces: YES Arithmetic: YES 1. m115(5,2): pr inv Volume: 3.060334984516746301926173678 Chern-Simons: 1.062061273492872714000000000 E-58 1. m115(5,2): pr eta inv Eta invariant: 1.504586804958427150458871723 E-58 3*eta - 2*CS: 0.E-76 1. m115(5,2): com inv Invariant trace field: x^10 - 2*x^9 + 3*x^8 - 3*x^7 - x^6 - x^4 - 3*x^3 + 3*x^2 - 2*x + 1 [2, 4] 9329048569 R(6) = 0.340186 + 0.940358*I 1. m115(5,2): pr arith Invariant trace field: x^10 - 2*x^9 + 3*x^8 - 3*x^7 - x^6 - x^4 - 3*x^3 + 3*x^2- 2*x + 1 [2, 4] 9329048569 R(6) = 0.340186 + 0.940358*I Invariant quaternion algebra: Hilbert Symbol: (1/8*x^9 + 9/8*x^8 - 5/4*x^7 + 15/8*x^6 - 5/2*x^5 - 7/2*x^4 - 13/8*x^3 - 9/4*x^2 - 19/8*x - 3/8, -3*x^8 + 5*x^7 - 5*x^6 + 4*x^5 + 8*x^4 - x^3- 3*x^2 + 7*x - 7) Real ramification: [1, 2] Finite ramification: [11, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0]~, 1, 2, [1, -3, 5, -5, -1, -5, 5, -3, 1,0]~] [11, [1, -5, 1, 0, 0, 0, 0, 0, 0, 0]~, 1, 2, [1, 3, -5, 2, 3, 2, -5, 3, 1, 0]~] Integer traces: YES Arithmetic: NO Borel Regulator: not computed (shape field required) 1. m121(-4,3): pr inv Volume: 3.195780918694216090819937082 Chern-Simons: -5.144359296365210477107454689 E-59 1. m121(-4,3): pr eta inv Eta invariant: -3.429572864243473651621810227 E-59 3*eta - 2*CS: 0.E-77 1. m121(-4,3): com inv Invariant trace field: x^8 - x^6 - 2*x^5 + 3*x^4 - 2*x^3 - x^2 + 1 [2, 3] -8454607 R(5) = 0.564069 + 0.825727*I 1. m121(-4,3): pr arith Invariant trace field: x^8 - x^6 - 2*x^5 + 3*x^4 - 2*x^3 - x^2 + 1 [2, 3] -8454607 R(5) = 0.564069 + 0.825727*I Invariant quaternion algebra: Hilbert Symbol: (-1/2*x^7 + x^6 + 1/2*x^5 - 2*x^3 + 7/2*x^2 - x - 9/2, 7*x^7 + 9/2*x^6 - 7*x^5 - 23*x^4 + 5*x^3 + 4*x^2 + 7/2*x) Real ramification: [1, 2] Finite ramification: [193, [-21, 1, 0, 0, 0, 0, 0, 0]~, 1, 1, [-74, 12, 7, 62, -52, -85, 42, 2]~] [193, [-46, 1, 0, 0, 0, 0, 0, 0]~, 1, 1, [-29, 76, 7, -49, 32, -16, 92, 2]~] Integer traces: YES Arithmetic: NO Borel Regulator: not computed (shape field required) 1. s942(-2,1): pr inv Volume: 3.663862376708876060218414059 Chern-Simons: -7.965459555662261385139411018 E-59 1. s942(-2,1): pr eta inv Eta invariant: -2.655153185220753794901910049 E-59 3*eta - 2*CS: -1.000000000000000000000000000 1. s942(-2,1): com inv Invariant trace field: x^2 + 1 [0, 1] -4 R(1) = 0.000000 + 1.00000*I 1. s942(-2,1): pr arith Invariant trace field: x^2 + 1 [0, 1] -4 R(1) = 0.000000 + 1.00000*I Invariant quaternion algebra: Hilbert Symbol: (8*x - 4, 48*x) Real ramification: [] Finite ramification: [2, [1, 1]~, 2, 1, [1, 1]~] [3, [3, 0]~, 1, 2, [1, 0]~] Integer traces: YES Arithmetic: YES Borel Regulator: not computed (shape field required) 1. m336(-1,3): pr inv Volume: 3.663862376708876060218414059 Chern-Simons: -5.310306370441507589803820099 E-59 1. m336(-1,3): pr eta inv Eta invariant: -6.859145727008581161000000000 E-59 3*eta - 2*CS: -1.000000000000000000000000000 1. m336(-1,3): com inv Invariant trace field: x^2 + 1 [0, 1] -4 R(1) = 0.000000 + 1.00000*I 1. m336(-1,3): pr arith Invariant trace field: x^2 + 1 [0, 1] -4 R(1) = 0.000000 + 1.00000*I Invariant quaternion algebra: Hilbert Symbol: (2*x - 4, 24*x - 24) Real ramification: [] Finite ramification: [2, [1, 1]~, 2, 1, [1, 1]~] [3, [3, 0]~, 1, 2, [1, 0]~] Integer traces: YES Arithmetic: YES Borel Regulator: not computed (shape field required) 1. m303(-1,3): pr inv Volume: 3.663862376708876060218414059 Chern-Simons: 2.655153184197843074000000000 E-59 1. m303(-1,3): pr eta inv Eta invariant: 2.101996271633096754170855202 E-59 3*eta - 2*CS: 0.E-77 1. m303(-1,3): com inv Invariant trace field: x^2 + 1 [0, 1] -4 R(1) = 0.000000 + 1.00000*I 1. m303(-1,3): pr arith Invariant trace field: x^2 + 1 [0, 1] -4 R(1) = 0.000000 + 1.00000*I Invariant quaternion algebra: Hilbert Symbol: (2*x - 4, -14) Real ramification: [] Finite ramification: [2, [1, 1]~, 2, 1, [1, 1]~] [7, [7, 0]~, 1, 2, [1, 0]~] Integer traces: YES Arithmetic: YES Borel Regulator: not computed (shape field required) 1. s960(-1,2): pr inv Volume: 3.663862376708876060218414059 Chern-Simons: 0.E-58 1. s960(-1,2): pr eta inv Eta invariant: 2.655153183984351744818993534 E-59 3*eta - 2*CS: 1.000000000000000000000000000 1. s960(-1,2): com inv Invariant trace field: x^2 + 1 [0, 1] -4 R(1) = 0.000000 + 1.00000*I 1. s960(-1,2): pr arith Invariant trace field: x^2 + 1 [0, 1] -4 R(1) = 0.000000 + 1.00000*I Invariant quaternion algebra: Hilbert Symbol: (8*x - 4, 24) Real ramification: [] Finite ramification: [2, [1, 1]~, 2, 1, [1, 1]~] [3, [3, 0]~, 1, 2, [1, 0]~] Integer traces: YES Arithmetic: YES Borel Regulator: not computed (shape field required) 1. s235(-3,4): pr inv Volume: 3.794090111820813358692493965 Chern-Simons: -5.310306370441507589803820099 E-59 1. s235(-3,4): pr eta inv Eta invariant: 3.097678715363144874000000000 E-59 3*eta - 2*CS: 1.000000000000000000000000000 1. s235(-3,4): com inv Invariant trace field not found. 1. m290(-3,4): pr inv Volume: 3.818259814789777436694741649 Chern-Simons: -1.062061273492872714000000000 E-58 1. m290(-3,4): pr eta inv Eta invariant: 1.548839358045439713656640789 E-59 3*eta - 2*CS: 0.E-77 1. m290(-3,4): com inv Invariant trace field: x^6 - x^5 - x^4 - x^2 - x + 1 [2, 2] 399424 R(3) = -0.906803 + 0.421554*I 1. m290(-3,4): pr arith Invariant trace field: x^6 - x^5 - x^4 - x^2 - x + 1 [2, 2] 399424 R(3) = -0.906803 + 0.421554*I Invariant quaternion algebra: Hilbert Symbol: (x^4 - 2*x^2 - 3, -7*x^5 + 10*x^4 + 4*x^3 - 6*x^2 + 17*x - 7) Real ramification: [] Finite ramification: [2, [3, 1, 1, 0, 0, 0]~, 1, 2, [1, 0, 0, 0, 1, 0]~] [31, [1, 4, 1, 0, 0, 0]~, 1, 2, [1, -5, -13, -5, 1, 0]~] Integer traces: NO Arithmetic: NO Borel Regulator: not computed (shape field required) 1. s287(3,4): pr inv Volume: 3.896345046488562840688673379 Chern-Simons: 1.593091911132452276962830073 E-58 1. s287(3,4): pr eta inv Eta invariant: 3.540204248609541126929095566 E-59 3*eta - 2*CS: 1.000000000000000000000000000 1. s287(3,4): com inv Invariant trace field not found. 1. m346(2,3): pr inv Volume: 3.933297923477320044254895280 Chern-Simons: 1.593091911132452276962830073 E-58 1. m346(2,3): pr eta inv Eta invariant: 5.974094669219500138793860211 E-59 3*eta - 2*CS: 0.E-77 1. m346(2,3): com inv Invariant trace field not found. 1. s912(0,1): pr inv Volume: 4.059766425638614500084810217 Chern-Simons: 1.062061273492872714000000000 E-58 1. s912(0,1): pr eta inv Eta invariant: -8.850510615855455398000000000 E-60 3*eta - 2*CS: 0.E-68 1. s912(0,1): com inv Invariant trace field: x^2 - x + 1 [0, 1] -3 R(1) = 0.500000 + 0.866025*I 1. s912(0,1): pr arith Invariant trace field: x^2 - x + 1 [0, 1] -3 R(1) = 0.500000 + 0.866025*I Invariant quaternion algebra: Hilbert Symbol: (5, 15*x + 15) Real ramification: [] Finite ramification: [3, [1, 1]~, 2, 1, [1, 1]~] [5, [5, 0]~, 1, 2, [1, 0]~] Integer traces: YES Arithmetic: YES Borel Regulator: not computed (shape field required) 1. m401(-2,3): pr inv Volume: 4.059766425638614500084810217 Chern-Simons: 0.E-58 1. m401(-2,3): pr eta inv Eta invariant: 4.646518066525459289000000000 E-59 3*eta - 2*CS: 0.E-67 1. m401(-2,3): com inv Invariant trace field: x^2 - x + 1 [0, 1] -3 R(1) = 0.500000 + 0.866025*I 1. m401(-2,3): pr arith Invariant trace field: x^2 - x + 1 [0, 1] -3 R(1) = 0.500000 + 0.866025*I Invariant quaternion algebra: Hilbert Symbol: (3*x - 4, -5*x - 5) Real ramification: [] Finite ramification: [3, [1, 1]~, 2, 1, [1, 1]~] [5, [5, 0]~, 1, 2, [1, 0]~] Integer traces: YES Arithmetic: YES Borel Regulator: not computed (shape field required) 1. v0825(4,1): pr inv Volume: 4.059766425638614500084810217 Chern-Simons: 0.E-58 1. v0825(4,1): pr eta inv Eta invariant: 0.6666666666666666666666666666 NONINTEGRAL ETA 3*eta - 2*CS: 2.000000000000000000000000000 1. v0825(4,1): com inv Invariant trace field: x^2 - x + 1 [0, 1] -3 R(1) = 0.500000 + 0.866025*I 1. v0825(4,1): pr arith Invariant trace field: x^2 - x + 1 [0, 1] -3 R(1) = 0.500000 + 0.866025*I Invariant quaternion algebra: Hilbert Symbol: (-x - 4, 18*x + 30) Real ramification: [] Finite ramification: [2, [2, 0]~, 1, 2, [1, 0]~] [3, [1, 1]~, 2, 1, [1, 1]~] Integer traces: YES Arithmetic: YES Borel Regulator: not computed (shape field required) 1. m358(1,3): pr inv Volume: 4.059766425638614500084810217 Chern-Simons: 7.965459555662261385139411018 E-59 1. m358(1,3): pr eta inv Eta invariant: 2.682811035845744847836216728 E-59 3*eta - 2*CS: 0.E-77 1. m358(1,3): com inv Invariant trace field: x^2 - x + 1 [0, 1] -3 R(1) = 0.500000 + 0.866025*I 1. m358(1,3): pr arith Invariant trace field: x^2 - x + 1 [0, 1] -3 R(1) = 0.500000 + 0.866025*I Invariant quaternion algebra: Hilbert Symbol: (x - 5, -24) Real ramification: [] Finite ramification: [2, [2, 0]~, 1, 2, [1, 0]~] [3, [1, 1]~, 2, 1, [1, 1]~] Integer traces: YES Arithmetic: YES Borel Regulator: not computed (shape field required) 1. s566(2,3): pr inv Volume: 4.116968736386066724912101479 Chern-Simons: 0.E-58 1. s566(2,3): pr eta inv Eta invariant: -0.6666666666666666666666666666 NONINTEGRAL ETA 3*eta - 2*CS: -3.000000000000000000000000000 1. s566(2,3): com inv Invariant trace field: x^4 - 2*x^3 + x^2 - 2*x + 1 [2, 1] -448 R(3) = -0.207106 + 0.978318*I 1. s566(2,3): pr arith Invariant trace field: x^4 - 2*x^3 + x^2 - 2*x + 1 [2, 1] -448 R(3) = -0.207106 + 0.978318*I Invariant quaternion algebra: Hilbert Symbol: (-x^2 + 2*x - 4, -2*x^3 + 10*x^2 - 16*x + 7) Real ramification: [2] Finite ramification: [7, [1, 1, 0, 0]~, 2, 1, [1, -3, -3, 1]~] Integer traces: YES Arithmetic: NO Borel Regulator: not computed (shape field required) 1. v2678(2,1): pr inv Volume: 4.116968736386066724912101479 Chern-Simons: 1.062061273492872714000000000 E-58 1. v2678(2,1): pr eta inv Eta invariant: -2.876415949419414560921104406 E-59 3*eta - 2*CS: 0.E-77 1. v2678(2,1): com inv Invariant trace field: x^4 - 2*x^3 + x^2 - 2*x + 1 [2, 1] -448 R(3) = -0.207106 + 0.978318*I 1. v2678(2,1): pr arith Invariant trace field: x^4 - 2*x^3 + x^2 - 2*x + 1 [2, 1] -448 R(3) = -0.207106 + 0.978318*I Invariant quaternion algebra: Hilbert Symbol: (-2*x^3 + x^2 + 6*x - 4, -8*x^2 + 19*x - 8) Real ramification: [1, 2] Finite ramification: none Integer traces: YES Arithmetic: YES Borel Regulator: not computed (shape field required) 1. s500(4,1): pr inv Volume: 4.116968736386066724912101479 Chern-Simons: 1.062061273492872714000000000 E-58 1. s500(4,1): pr eta inv Eta invariant: 4.508228845739404880895578875 E-59 3*eta - 2*CS: 0.E-77 1. s500(4,1): com inv Invariant trace field: x^4 - 2*x^3 + x^2 - 2*x + 1 [2, 1] -448 R(3) = -0.207106 + 0.978318*I 1. s500(4,1): pr arith Invariant trace field: x^4 - 2*x^3 + x^2 - 2*x + 1 [2, 1] -448 R(3) = -0.207106 + 0.978318*I Invariant quaternion algebra: Hilbert Symbol: (x^2 - 4, 6*x^3 - 2*x^2 - 24*x + 12) Real ramification: [1, 2] Finite ramification: none Integer traces: YES Arithmetic: YES Borel Regulator: not computed (shape field required) 1. s668(4,1): pr inv Volume: 4.221804877303242915770762629 Chern-Simons: 1.062061273492872714000000000 E-58 1. s668(4,1): pr eta inv Eta invariant: -4.701833765495084845129764028 E-59 3*eta - 2*CS: 0.E-77 1. s668(4,1): com inv Invariant trace field not found. 1. v2817(-3,1): pr inv Volume: 4.407345721058114044128986726 Chern-Simons: -5.310306368395686149000000000 E-59 1. v2817(-3,1): pr eta inv Eta invariant: -1.714786432121736825810905113 E-59 3*eta - 2*CS: 0.E-77 1. v2817(-3,1): com inv Invariant trace field not found. 1. s533(1,4): pr inv Volume: 4.422687775755177490059127725 Chern-Simons: 3.186183821409940719000000000 E-58 1. s533(1,4): pr eta inv Eta invariant: 1.637344464878879266868903774 E-58 3*eta - 2*CS: 0.E-76 1. s533(1,4): com inv Invariant trace field not found. 1. m402(2,3): pr inv Volume: 4.436783913348955986006967526 Chern-Simons: 5.310306368395686149000000000 E-59 1. m402(2,3): pr eta inv Eta invariant: -4.259308232152155112431434028 E-59 3*eta - 2*CS: 0.E-86 1. m402(2,3): com inv Invariant trace field not found. 1. v2585(1,2): pr inv Volume: 4.465424526771380877096900272 Chern-Simons: 0.E-58 1. v2585(1,2): pr eta inv Eta invariant: -2.655153182335197925000000000 E-59 3*eta - 2*CS: -1.000000000000000000000000000 1. v2585(1,2): com inv Invariant trace field: x^8 - 3*x^7 + 3*x^6 + 3*x^5 - 7*x^4 + 3*x^3 + 3*x^2 - 3*x + 1 [0, 4] 4626801 R(3) = 0.821965 + 0.569538*I 1. v2585(1,2): pr arith Invariant trace field: x^8 - 3*x^7 + 3*x^6 + 3*x^5 - 7*x^4 + 3*x^3 + 3*x^2 - 3*x + 1 [0, 4] 4626801 R(3) = 0.821965 + 0.569538*I Invariant quaternion algebra: Hilbert Symbol: (1/4*x^7 + 2*x^6 - 7/2*x^5 + 9/4*x^4 + 13/2*x^3 + 1/4*x - 2, 105/8*x^7 - 315/8*x^6 + 105/4*x^5 + 459/8*x^4 - 705/8*x^3 - 59/4*x^2 + 345/8*x - 171/8) Real ramification: [] WARNING: ramification test aborted Finite ramification: [-2, [1, 0, 0, 1, 0, 0, 1, 0]~, 1, 6, [1, 1, 1, 0, 0, 0, 0, 0]~] [17, [1, -4, 1, 0, 0, 0, 0, 0]~, 1, 2, [1, 1, 6, -8, 6, 1, 1, 0]~] Integer traces: NO Arithmetic: NO Borel Regulator: not computed (shape field required) 1. v1461(1,3): pr inv Volume: 4.598034401324609701701012907 Chern-Simons: 0.E-58 1. v1461(1,3): pr eta inv Eta invariant: -6.637882912531495094000000000 E-60 3*eta - 2*CS: 0.E-68 1. v1461(1,3): com inv Invariant trace field: x^10 - 8*x^8 + 14*x^6 + 27*x^4 - 41*x^2 + 3 [4, 3] -10464371979312 R(6) = 0.000000 + 1.30573*I 1. v1461(1,3): pr arith Invariant trace field: x^10 - 8*x^8 + 14*x^6 + 27*x^4 - 41*x^2 + 3 [4, 3] -10464371979312 R(6) = 0.000000 + 1.30573*I Invariant quaternion algebra: Hilbert Symbol: (-1/24*x^9 - 1/18*x^8 + 37/120*x^7 + 37/90*x^6 - 11/24*x^5 - 11/18*x^4 - x^3 - 7/6*x^2 + 169/120*x - 191/90, 27/40*x^9 + 11/8*x^8 - 199/40*x^7 - 411/40*x^6 + 57/8*x^5 +125/8*x^4 + 78/5*x^3 + 30*x^2 - 3/40*x - 127/40) Real ramification: [2, 3] WARNING: ramification test aborted Finite ramification: [2, [0, 1, 0, 0, 0, 1, 1, 1, 1, 1]~, 2, 2, [1, 2, 1, 1, 1, 0, 0, 0, 0, 0]~] [-2, [1, 1, 0, 0, 0, 1, 1, 1, 1, 1]~, 1, 6, [1, 2, 1, 0, 0, 0, 0, 0, 0, 0]~] Integer traces: NO Arithmetic: NO Borel Regulator: not computed (shape field required) 1. v1222(-5,1): pr inv Volume: 4.626243216569129885777945214 Chern-Simons: -7.965459555662261385139411018 E-59 1. v1222(-5,1): pr eta inv Eta invariant: -4.181866264744443946292651315 E-59 3*eta - 2*CS: 0.E-77 1. v1222(-5,1): com inv Invariant trace field: x^16 - 2*x^15 - x^12 + 5*x^11 - 5*x^10 + 7*x^9 - 6*x^8 + 7*x^7 - 5*x^6 + 5*x^5 - x^4 - 2*x + 1 [6, 5] -20714696261256045624976 R(7) = -0.685136 + 0.728415*I 1. v1222(-5,1): pr arith Invariant trace field: x^16 - 2*x^15 - x^12 + 5*x^11 - 5*x^10 + 7*x^9 - 6*x^8 + 7*x^7 - 5*x^6 + 5*x^5 - x^4 - 2*x + 1 [6, 5] -20714696261256045624976 R(7) = -0.685136 + 0.728415*I Invariant quaternion algebra: Hilbert Symbol: (11/6*x^15 - 2*x^14 - 3/2*x^13 - 3*x^12 - 10/3*x^11 + 17/2*x^10 - 19/6*x^9 + 11*x^8 - 9/2*x^7 + 71/6*x^6 - 2*x^5 + 29/3*x^4 + 7/2*x^3 + 4*x^2 + 3/2*x - 20/3, -15*x^15 + 20*x^14 + 15*x^13 + 6*x^12 + 23*x^11 - 65*x^10 + 28*x^9 - 69*x^8 + 19*x^7 - 63*x^6 + 17*x^5 - 53*x^4 - 13*x^3 - 17*x^2 - 2*x + 27) Real ramification: [1, 2, 4, 5] Finite ramification: [2, [3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]~, 1, 2, [0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0]~] [41, [1, -11, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]~, 1, 2, [0, 9, 15, 3, 16, 13, 9, -20, 9, 13, 16, 3, 15, 9, 2, 0]~] Integer traces: YES Arithmetic: NO Borel Regulator: not computed (shape field required) 1. v1251(4,3): pr inv Volume: 4.686034273802612530712490649 Chern-Simons: 1.062061273492872714000000000 E-58 1. v1251(4,3): pr eta inv Eta invariant: -1.548839355572635613319797240 E-59 3*eta - 2*CS: 0.E-87 1. v1251(4,3): com inv Invariant trace field: x^4 - x^3 + 2*x - 1 [2, 1] -275 R(3) = 0.809017 + 0.981593*I 1. v1251(4,3): pr arith Invariant trace field: x^4 - x^3 + 2*x - 1 [2, 1] -275 R(3) = 0.809017 + 0.981593*I Invariant quaternion algebra: Hilbert Symbol: (x^2 - x - 3, -18*x^3 - 18*x^2 + 7*x + 4) Real ramification: [1, 2] Finite ramification: [5, [-2, 2, 1, 0]~, 2, 2, [-2, 2, 1, 0]~] [11, [-2, 1, 0, 0]~, 2, 1, [-5, 2, 1, 1]~] Integer traces: YES Arithmetic: YES Borel Regulator: not computed (shape field required) 1. v1695(-5,1): pr inv Volume: 4.834441637051395316124910884 Chern-Simons: -1.593091911132452276962830073 E-58 1. v1695(-5,1): pr eta inv Eta invariant: -5.177548709202226619435077692 E-59 3*eta - 2*CS: -1.000000000000000000000000000 1. v1695(-5,1): com inv Invariant trace field not found. 1. v1860(2,3): pr inv Volume: 4.974542472997789726365829831 Chern-Simons: 1.062061273492872714000000000 E-58 1. v1860(2,3): pr eta inv Eta invariant: 5.089043607479248873971117150 E-59 3*eta - 2*CS: 0.E-77 1. v1860(2,3): com inv Invariant trace field not found. 1. v1847(-4,3): pr inv Volume: 5.016110016223202851478710196 Chern-Simons: -5.310306368395686149000000000 E-59 1. v1847(-4,3): pr eta inv Eta invariant: -6.527251581570755129338423650 E-59 3*eta - 2*CS: 0.E-77 1. v1847(-4,3): com inv Invariant trace field not found. 1. v1845(-5,2): pr inv Volume: 5.017640518843689544736089834 Chern-Simons: -1.858607229311019182000000000 E-58 1. v1845(-5,2): pr eta inv Eta invariant: -7.080408491194248199000000000 E-59 3*eta - 2*CS: 0.E-67 1. v1845(-5,2): com inv Invariant trace field not found. 1. v3178(-3,1): pr inv Volume: 5.112354279842939646718308536 Chern-Simons: 5.310306370441507589803820099 E-59 1. v3178(-3,1): pr eta inv Eta invariant: 2.212627653032541275000000000 E-58 3*eta - 2*CS: 0.E-66 1. v3178(-3,1): com inv Invariant trace field not found. 1. v3066(-1,2): pr inv Volume: 5.137941201873417769841348339 Chern-Simons: 0.E-58 1. v3066(-1,2): pr eta inv Eta invariant: 5.531569134443998336000000000 E-60 3*eta - 2*CS: 0.E-68 1. v3066(-1,2): com inv Invariant trace field: x^6 - 3*x^5 + 10*x^4 - 15*x^3 + 21*x^2 - 14*x + 4 [0, 3] -205379 R(1) = 0.273301 + 1.76942*I 1. v3066(-1,2): pr arith Invariant trace field: x^6 - 3*x^5 + 10*x^4 - 15*x^3 + 21*x^2 - 14*x + 4 [0, 3] -205379 R(1) = 0.273301 + 1.76942*I Invariant quaternion algebra: Hilbert Symbol: (1/2*x^4 - x^3 + 3*x^2 - 5/2*x + 5, -3/2*x^5 + 6*x^4 - 17*x^3 + 51/2*x^2 -28*x + 11) Real ramification: [] Finite ramification: [13, [-6, -3, 1, 0, 0, 0]~, 1, 2, [-5, 6, 3, 0, 2, 0]~] [59, [-16, 1, 0, 0, 0, 0]~, 2, 1, [-15, -2, -9, -18, 26, 2]~] Integer traces: YES Arithmetic: NO Borel Regulator: not computed (shape field required) 1. v3283(-3,1): pr inv Volume: 5.171469893437072862970586504 Chern-Simons: 0.E-58 1. v3283(-3,1): pr eta inv Eta invariant: 5.310306368395686149000000000 E-59 3*eta - 2*CS: 1.000000000000000000000000000 1. v3283(-3,1): com inv Invariant trace field not found. 1. v3347(3,1): pr inv Volume: 5.190775187373484276603987668 Chern-Simons: 3.186183821409940719000000000 E-58 1. v3347(3,1): pr eta inv Eta invariant: 0.3333333333333333333333333333 NONINTEGRAL ETA 3*eta - 2*CS: 1.000000000000000000000000000 1. v3347(3,1): com inv Invariant trace field: x^4 - x^3 - x^2 - x + 1 [2, 1] -507 R(3) = -0.651388 + 0.758745*I 1. v3347(3,1): pr arith Invariant trace field: x^4 - x^3 - x^2 - x + 1 [2, 1] -507 R(3) = -0.651388 + 0.758745*I Invariant quaternion algebra: Hilbert Symbol: (x^3 - 3*x^2 + x - 1, -8*x^3 + 13*x^2 + 2*x - 4) Real ramification: [1, 2] Finite ramification: none Integer traces: YES Arithmetic: YES Borel Regulator: not computed (shape field required) 1. v3390(3,1): pr inv Volume: 5.333489566898119581593424925 Chern-Simons: 1.062061273492872714000000000 E-58 1. v3390(3,1): pr eta inv Eta invariant: -0.3333333333333333333333333333 3*eta - 2*CS: -1.000000000000000000000000000 1. v3390(3,1): com inv Invariant trace field: x^2 - x + 2 [0, 1] -7 R(1) = 0.500000 + 1.32287*I 1. v3390(3,1): pr arith Invariant trace field: x^2 - x + 2 [0, 1] -7 R(1) = 0.500000 + 1.32287*I Invariant quaternion algebra: Hilbert Symbol: (3*x - 5, 42*x - 14) Real ramification: [] Finite ramification: [2, [0, 1]~, 1, 1, [1, 1]~] [7, [3, 1]~, 2, 1, [3, 1]~] Integer traces: NO Arithmetic: NO Borel Regulator: not computed (shape field required) 1. v3183(-3,2): pr inv Volume: 5.573609112831137043711018858 Chern-Simons: 0.E-58 1. v3183(-3,2): pr eta inv Eta invariant: -1.000000000000000000000000000 3*eta - 2*CS: -3.000000000000000000000000000 1. v3183(-3,2): com inv Invariant trace field: x^4 - x^3 - 2*x^2 - x + 1 [2, 1] -1156 R(3) = -0.780776 + 0.624810*I 1. v3183(-3,2): pr arith Invariant trace field: x^4 - x^3 - 2*x^2 - x + 1 [2, 1] -1156 R(3) = -0.780776 + 0.624810*I Invariant quaternion algebra: Hilbert Symbol: (x^3 - 2*x^2 - x - 2, x^2 - 4*x + 1) Real ramification: [1, 2] Finite ramification: none Integer traces: YES Arithmetic: YES Borel Regulator: not computed (shape field required) 1. v3439(3,2): pr inv Volume: 5.614494740828190882519428785 Chern-Simons: 1.062061273492872714000000000 E-58 1. v3439(3,2): pr eta inv Eta invariant: 3.540204245597124099000000000 E-59 3*eta - 2*CS: 0.E-67 1. v3439(3,2): com inv Invariant trace field not found.