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Properties

Label 32.0.54520681807...3009.1
Degree $32$
Signature $[0, 16]$
Discriminant $13^{16}\cdot 17^{30}$
Root discriminant $51.35$
Ramified primes $13, 17$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{16}$ (as 32T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43046721, 14348907, 19131876, 11160261, 10097379, 7085880, 5727753, 4271211, 3332988, 2534733, 1955907, 1496880, 1150929, 882603, 677844, 520149, 399331, -173383, 75316, -32689, 14209, -6160, 2683, -1159, 508, -217, 97, -40, 19, -7, 4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 + 4*x^30 - 7*x^29 + 19*x^28 - 40*x^27 + 97*x^26 - 217*x^25 + 508*x^24 - 1159*x^23 + 2683*x^22 - 6160*x^21 + 14209*x^20 - 32689*x^19 + 75316*x^18 - 173383*x^17 + 399331*x^16 + 520149*x^15 + 677844*x^14 + 882603*x^13 + 1150929*x^12 + 1496880*x^11 + 1955907*x^10 + 2534733*x^9 + 3332988*x^8 + 4271211*x^7 + 5727753*x^6 + 7085880*x^5 + 10097379*x^4 + 11160261*x^3 + 19131876*x^2 + 14348907*x + 43046721)
 
gp: K = bnfinit(x^32 - x^31 + 4*x^30 - 7*x^29 + 19*x^28 - 40*x^27 + 97*x^26 - 217*x^25 + 508*x^24 - 1159*x^23 + 2683*x^22 - 6160*x^21 + 14209*x^20 - 32689*x^19 + 75316*x^18 - 173383*x^17 + 399331*x^16 + 520149*x^15 + 677844*x^14 + 882603*x^13 + 1150929*x^12 + 1496880*x^11 + 1955907*x^10 + 2534733*x^9 + 3332988*x^8 + 4271211*x^7 + 5727753*x^6 + 7085880*x^5 + 10097379*x^4 + 11160261*x^3 + 19131876*x^2 + 14348907*x + 43046721, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} + 4 x^{30} - 7 x^{29} + 19 x^{28} - 40 x^{27} + 97 x^{26} - 217 x^{25} + 508 x^{24} - 1159 x^{23} + 2683 x^{22} - 6160 x^{21} + 14209 x^{20} - 32689 x^{19} + 75316 x^{18} - 173383 x^{17} + 399331 x^{16} + 520149 x^{15} + 677844 x^{14} + 882603 x^{13} + 1150929 x^{12} + 1496880 x^{11} + 1955907 x^{10} + 2534733 x^{9} + 3332988 x^{8} + 4271211 x^{7} + 5727753 x^{6} + 7085880 x^{5} + 10097379 x^{4} + 11160261 x^{3} + 19131876 x^{2} + 14348907 x + 43046721 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5452068180730262807665859440063268048455458111797023009=13^{16}\cdot 17^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.35$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(221=13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{221}(1,·)$, $\chi_{221}(131,·)$, $\chi_{221}(129,·)$, $\chi_{221}(12,·)$, $\chi_{221}(66,·)$, $\chi_{221}(142,·)$, $\chi_{221}(144,·)$, $\chi_{221}(25,·)$, $\chi_{221}(155,·)$, $\chi_{221}(157,·)$, $\chi_{221}(27,·)$, $\chi_{221}(38,·)$, $\chi_{221}(40,·)$, $\chi_{221}(92,·)$, $\chi_{221}(53,·)$, $\chi_{221}(183,·)$, $\chi_{221}(181,·)$, $\chi_{221}(64,·)$, $\chi_{221}(194,·)$, $\chi_{221}(196,·)$, $\chi_{221}(77,·)$, $\chi_{221}(79,·)$, $\chi_{221}(209,·)$, $\chi_{221}(14,·)$, $\chi_{221}(90,·)$, $\chi_{221}(207,·)$, $\chi_{221}(220,·)$, $\chi_{221}(103,·)$, $\chi_{221}(105,·)$, $\chi_{221}(168,·)$, $\chi_{221}(116,·)$, $\chi_{221}(118,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1197993} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{173383}{399331}$, $\frac{1}{3593979} a^{18} - \frac{1}{3593979} a^{17} + \frac{4}{9} a^{16} + \frac{2}{9} a^{15} + \frac{1}{9} a^{14} - \frac{4}{9} a^{13} - \frac{2}{9} a^{12} - \frac{1}{9} a^{11} + \frac{4}{9} a^{10} + \frac{2}{9} a^{9} + \frac{1}{9} a^{8} - \frac{4}{9} a^{7} - \frac{2}{9} a^{6} - \frac{1}{9} a^{5} + \frac{4}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2} + \frac{173383}{1197993} a + \frac{75316}{399331}$, $\frac{1}{10781937} a^{19} - \frac{1}{10781937} a^{18} + \frac{4}{10781937} a^{17} - \frac{7}{27} a^{16} - \frac{8}{27} a^{15} - \frac{13}{27} a^{14} - \frac{11}{27} a^{13} - \frac{1}{27} a^{12} - \frac{5}{27} a^{11} + \frac{2}{27} a^{10} + \frac{10}{27} a^{9} - \frac{4}{27} a^{8} + \frac{7}{27} a^{7} + \frac{8}{27} a^{6} + \frac{13}{27} a^{5} + \frac{11}{27} a^{4} + \frac{1}{27} a^{3} + \frac{173383}{3593979} a^{2} + \frac{75316}{1197993} a + \frac{32689}{399331}$, $\frac{1}{32345811} a^{20} - \frac{1}{32345811} a^{19} + \frac{4}{32345811} a^{18} - \frac{7}{32345811} a^{17} + \frac{19}{81} a^{16} - \frac{40}{81} a^{15} + \frac{16}{81} a^{14} + \frac{26}{81} a^{13} + \frac{22}{81} a^{12} - \frac{25}{81} a^{11} + \frac{10}{81} a^{10} - \frac{4}{81} a^{9} + \frac{34}{81} a^{8} + \frac{35}{81} a^{7} - \frac{14}{81} a^{6} + \frac{38}{81} a^{5} + \frac{1}{81} a^{4} + \frac{173383}{10781937} a^{3} + \frac{75316}{3593979} a^{2} + \frac{32689}{1197993} a + \frac{14209}{399331}$, $\frac{1}{97037433} a^{21} - \frac{1}{97037433} a^{20} + \frac{4}{97037433} a^{19} - \frac{7}{97037433} a^{18} + \frac{19}{97037433} a^{17} + \frac{41}{243} a^{16} + \frac{16}{243} a^{15} + \frac{107}{243} a^{14} - \frac{59}{243} a^{13} - \frac{106}{243} a^{12} - \frac{71}{243} a^{11} - \frac{4}{243} a^{10} + \frac{34}{243} a^{9} - \frac{46}{243} a^{8} - \frac{95}{243} a^{7} - \frac{43}{243} a^{6} + \frac{1}{243} a^{5} + \frac{173383}{32345811} a^{4} + \frac{75316}{10781937} a^{3} + \frac{32689}{3593979} a^{2} + \frac{14209}{1197993} a + \frac{6160}{399331}$, $\frac{1}{291112299} a^{22} - \frac{1}{291112299} a^{21} + \frac{4}{291112299} a^{20} - \frac{7}{291112299} a^{19} + \frac{19}{291112299} a^{18} - \frac{40}{291112299} a^{17} - \frac{227}{729} a^{16} + \frac{350}{729} a^{15} - \frac{302}{729} a^{14} - \frac{106}{729} a^{13} - \frac{71}{729} a^{12} - \frac{247}{729} a^{11} + \frac{34}{729} a^{10} - \frac{46}{729} a^{9} + \frac{148}{729} a^{8} - \frac{286}{729} a^{7} + \frac{1}{729} a^{6} + \frac{173383}{97037433} a^{5} + \frac{75316}{32345811} a^{4} + \frac{32689}{10781937} a^{3} + \frac{14209}{3593979} a^{2} + \frac{6160}{1197993} a + \frac{2683}{399331}$, $\frac{1}{873336897} a^{23} - \frac{1}{873336897} a^{22} + \frac{4}{873336897} a^{21} - \frac{7}{873336897} a^{20} + \frac{19}{873336897} a^{19} - \frac{40}{873336897} a^{18} + \frac{97}{873336897} a^{17} + \frac{1079}{2187} a^{16} + \frac{427}{2187} a^{15} + \frac{623}{2187} a^{14} + \frac{658}{2187} a^{13} - \frac{976}{2187} a^{12} + \frac{763}{2187} a^{11} + \frac{683}{2187} a^{10} - \frac{581}{2187} a^{9} + \frac{443}{2187} a^{8} + \frac{1}{2187} a^{7} + \frac{173383}{291112299} a^{6} + \frac{75316}{97037433} a^{5} + \frac{32689}{32345811} a^{4} + \frac{14209}{10781937} a^{3} + \frac{6160}{3593979} a^{2} + \frac{2683}{1197993} a + \frac{1159}{399331}$, $\frac{1}{2620010691} a^{24} - \frac{1}{2620010691} a^{23} + \frac{4}{2620010691} a^{22} - \frac{7}{2620010691} a^{21} + \frac{19}{2620010691} a^{20} - \frac{40}{2620010691} a^{19} + \frac{97}{2620010691} a^{18} - \frac{217}{2620010691} a^{17} + \frac{427}{6561} a^{16} + \frac{2810}{6561} a^{15} - \frac{1529}{6561} a^{14} - \frac{3163}{6561} a^{13} - \frac{1424}{6561} a^{12} - \frac{1504}{6561} a^{11} - \frac{2768}{6561} a^{10} - \frac{1744}{6561} a^{9} + \frac{1}{6561} a^{8} + \frac{173383}{873336897} a^{7} + \frac{75316}{291112299} a^{6} + \frac{32689}{97037433} a^{5} + \frac{14209}{32345811} a^{4} + \frac{6160}{10781937} a^{3} + \frac{2683}{3593979} a^{2} + \frac{1159}{1197993} a + \frac{508}{399331}$, $\frac{1}{7860032073} a^{25} - \frac{1}{7860032073} a^{24} + \frac{4}{7860032073} a^{23} - \frac{7}{7860032073} a^{22} + \frac{19}{7860032073} a^{21} - \frac{40}{7860032073} a^{20} + \frac{97}{7860032073} a^{19} - \frac{217}{7860032073} a^{18} + \frac{508}{7860032073} a^{17} + \frac{9371}{19683} a^{16} - \frac{8090}{19683} a^{15} - \frac{3163}{19683} a^{14} - \frac{1424}{19683} a^{13} - \frac{8065}{19683} a^{12} + \frac{3793}{19683} a^{11} - \frac{8305}{19683} a^{10} + \frac{1}{19683} a^{9} + \frac{173383}{2620010691} a^{8} + \frac{75316}{873336897} a^{7} + \frac{32689}{291112299} a^{6} + \frac{14209}{97037433} a^{5} + \frac{6160}{32345811} a^{4} + \frac{2683}{10781937} a^{3} + \frac{1159}{3593979} a^{2} + \frac{508}{1197993} a + \frac{217}{399331}$, $\frac{1}{23580096219} a^{26} - \frac{1}{23580096219} a^{25} + \frac{4}{23580096219} a^{24} - \frac{7}{23580096219} a^{23} + \frac{19}{23580096219} a^{22} - \frac{40}{23580096219} a^{21} + \frac{97}{23580096219} a^{20} - \frac{217}{23580096219} a^{19} + \frac{508}{23580096219} a^{18} - \frac{1159}{23580096219} a^{17} + \frac{11593}{59049} a^{16} + \frac{16520}{59049} a^{15} + \frac{18259}{59049} a^{14} - \frac{27748}{59049} a^{13} + \frac{23476}{59049} a^{12} + \frac{11378}{59049} a^{11} + \frac{1}{59049} a^{10} + \frac{173383}{7860032073} a^{9} + \frac{75316}{2620010691} a^{8} + \frac{32689}{873336897} a^{7} + \frac{14209}{291112299} a^{6} + \frac{6160}{97037433} a^{5} + \frac{2683}{32345811} a^{4} + \frac{1159}{10781937} a^{3} + \frac{508}{3593979} a^{2} + \frac{217}{1197993} a + \frac{97}{399331}$, $\frac{1}{70740288657} a^{27} - \frac{1}{70740288657} a^{26} + \frac{4}{70740288657} a^{25} - \frac{7}{70740288657} a^{24} + \frac{19}{70740288657} a^{23} - \frac{40}{70740288657} a^{22} + \frac{97}{70740288657} a^{21} - \frac{217}{70740288657} a^{20} + \frac{508}{70740288657} a^{19} - \frac{1159}{70740288657} a^{18} + \frac{2683}{70740288657} a^{17} + \frac{16520}{177147} a^{16} + \frac{18259}{177147} a^{15} + \frac{31301}{177147} a^{14} + \frac{23476}{177147} a^{13} + \frac{70427}{177147} a^{12} + \frac{1}{177147} a^{11} + \frac{173383}{23580096219} a^{10} + \frac{75316}{7860032073} a^{9} + \frac{32689}{2620010691} a^{8} + \frac{14209}{873336897} a^{7} + \frac{6160}{291112299} a^{6} + \frac{2683}{97037433} a^{5} + \frac{1159}{32345811} a^{4} + \frac{508}{10781937} a^{3} + \frac{217}{3593979} a^{2} + \frac{97}{1197993} a + \frac{40}{399331}$, $\frac{1}{212220865971} a^{28} - \frac{1}{212220865971} a^{27} + \frac{4}{212220865971} a^{26} - \frac{7}{212220865971} a^{25} + \frac{19}{212220865971} a^{24} - \frac{40}{212220865971} a^{23} + \frac{97}{212220865971} a^{22} - \frac{217}{212220865971} a^{21} + \frac{508}{212220865971} a^{20} - \frac{1159}{212220865971} a^{19} + \frac{2683}{212220865971} a^{18} - \frac{6160}{212220865971} a^{17} + \frac{18259}{531441} a^{16} + \frac{31301}{531441} a^{15} + \frac{23476}{531441} a^{14} + \frac{70427}{531441} a^{13} + \frac{1}{531441} a^{12} + \frac{173383}{70740288657} a^{11} + \frac{75316}{23580096219} a^{10} + \frac{32689}{7860032073} a^{9} + \frac{14209}{2620010691} a^{8} + \frac{6160}{873336897} a^{7} + \frac{2683}{291112299} a^{6} + \frac{1159}{97037433} a^{5} + \frac{508}{32345811} a^{4} + \frac{217}{10781937} a^{3} + \frac{97}{3593979} a^{2} + \frac{40}{1197993} a + \frac{19}{399331}$, $\frac{1}{636662597913} a^{29} - \frac{1}{636662597913} a^{28} + \frac{4}{636662597913} a^{27} - \frac{7}{636662597913} a^{26} + \frac{19}{636662597913} a^{25} - \frac{40}{636662597913} a^{24} + \frac{97}{636662597913} a^{23} - \frac{217}{636662597913} a^{22} + \frac{508}{636662597913} a^{21} - \frac{1159}{636662597913} a^{20} + \frac{2683}{636662597913} a^{19} - \frac{6160}{636662597913} a^{18} + \frac{14209}{636662597913} a^{17} + \frac{31301}{1594323} a^{16} + \frac{23476}{1594323} a^{15} + \frac{70427}{1594323} a^{14} + \frac{1}{1594323} a^{13} + \frac{173383}{212220865971} a^{12} + \frac{75316}{70740288657} a^{11} + \frac{32689}{23580096219} a^{10} + \frac{14209}{7860032073} a^{9} + \frac{6160}{2620010691} a^{8} + \frac{2683}{873336897} a^{7} + \frac{1159}{291112299} a^{6} + \frac{508}{97037433} a^{5} + \frac{217}{32345811} a^{4} + \frac{97}{10781937} a^{3} + \frac{40}{3593979} a^{2} + \frac{19}{1197993} a + \frac{7}{399331}$, $\frac{1}{1909987793739} a^{30} - \frac{1}{1909987793739} a^{29} + \frac{4}{1909987793739} a^{28} - \frac{7}{1909987793739} a^{27} + \frac{19}{1909987793739} a^{26} - \frac{40}{1909987793739} a^{25} + \frac{97}{1909987793739} a^{24} - \frac{217}{1909987793739} a^{23} + \frac{508}{1909987793739} a^{22} - \frac{1159}{1909987793739} a^{21} + \frac{2683}{1909987793739} a^{20} - \frac{6160}{1909987793739} a^{19} + \frac{14209}{1909987793739} a^{18} - \frac{32689}{1909987793739} a^{17} + \frac{23476}{4782969} a^{16} + \frac{70427}{4782969} a^{15} + \frac{1}{4782969} a^{14} + \frac{173383}{636662597913} a^{13} + \frac{75316}{212220865971} a^{12} + \frac{32689}{70740288657} a^{11} + \frac{14209}{23580096219} a^{10} + \frac{6160}{7860032073} a^{9} + \frac{2683}{2620010691} a^{8} + \frac{1159}{873336897} a^{7} + \frac{508}{291112299} a^{6} + \frac{217}{97037433} a^{5} + \frac{97}{32345811} a^{4} + \frac{40}{10781937} a^{3} + \frac{19}{3593979} a^{2} + \frac{7}{1197993} a + \frac{4}{399331}$, $\frac{1}{5729963381217} a^{31} - \frac{1}{5729963381217} a^{30} + \frac{4}{5729963381217} a^{29} - \frac{7}{5729963381217} a^{28} + \frac{19}{5729963381217} a^{27} - \frac{40}{5729963381217} a^{26} + \frac{97}{5729963381217} a^{25} - \frac{217}{5729963381217} a^{24} + \frac{508}{5729963381217} a^{23} - \frac{1159}{5729963381217} a^{22} + \frac{2683}{5729963381217} a^{21} - \frac{6160}{5729963381217} a^{20} + \frac{14209}{5729963381217} a^{19} - \frac{32689}{5729963381217} a^{18} + \frac{75316}{5729963381217} a^{17} + \frac{70427}{14348907} a^{16} + \frac{1}{14348907} a^{15} + \frac{173383}{1909987793739} a^{14} + \frac{75316}{636662597913} a^{13} + \frac{32689}{212220865971} a^{12} + \frac{14209}{70740288657} a^{11} + \frac{6160}{23580096219} a^{10} + \frac{2683}{7860032073} a^{9} + \frac{1159}{2620010691} a^{8} + \frac{508}{873336897} a^{7} + \frac{217}{291112299} a^{6} + \frac{97}{97037433} a^{5} + \frac{40}{32345811} a^{4} + \frac{19}{10781937} a^{3} + \frac{7}{3593979} a^{2} + \frac{4}{1197993} a + \frac{1}{399331}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{508}{7860032073} a^{26} - \frac{727060321}{7860032073} a^{9} \) (order $34$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{16}$ (as 32T32):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2\times C_{16}$
Character table for $C_2\times C_{16}$ is not computed

Intermediate fields

\(\Q(\sqrt{221}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{13}, \sqrt{17})\), 4.4.4913.1, 4.4.830297.1, 8.8.689393108209.1, \(\Q(\zeta_{17})^+\), 8.8.11719682839553.1, 16.16.137350965859713069141239809.1, \(\Q(\zeta_{17})\), 16.0.2334966419615122175401076753.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{4}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17Data not computed