Properties

Label 31.31.107...849.1
Degree $31$
Signature $[31, 0]$
Discriminant $1.078\times 10^{85}$
Root discriminant $553.33$
Ramified prime $683$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{31}$ (as 31T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^31 - x^30 - 330*x^29 + 852*x^28 + 43784*x^27 - 173944*x^26 - 2923572*x^25 + 15562316*x^24 + 100635650*x^23 - 707709554*x^22 - 1616621784*x^21 + 17266725972*x^20 + 7046247776*x^19 - 243865756432*x^18 + 142454592564*x^17 + 2084163747748*x^16 - 2521472241195*x^15 - 10836712986085*x^14 + 18499228115810*x^13 + 32698257976552*x^12 - 74538290301632*x^11 - 47884603550112*x^10 + 168256221920320*x^9 + 4878741111552*x^8 - 195294436805376*x^7 + 62882730299136*x^6 + 92885896276480*x^5 - 43440624908288*x^4 - 15171034030080*x^3 + 6823763935232*x^2 + 786648006656*x + 5192548352)
 
gp: K = bnfinit(x^31 - x^30 - 330*x^29 + 852*x^28 + 43784*x^27 - 173944*x^26 - 2923572*x^25 + 15562316*x^24 + 100635650*x^23 - 707709554*x^22 - 1616621784*x^21 + 17266725972*x^20 + 7046247776*x^19 - 243865756432*x^18 + 142454592564*x^17 + 2084163747748*x^16 - 2521472241195*x^15 - 10836712986085*x^14 + 18499228115810*x^13 + 32698257976552*x^12 - 74538290301632*x^11 - 47884603550112*x^10 + 168256221920320*x^9 + 4878741111552*x^8 - 195294436805376*x^7 + 62882730299136*x^6 + 92885896276480*x^5 - 43440624908288*x^4 - 15171034030080*x^3 + 6823763935232*x^2 + 786648006656*x + 5192548352, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5192548352, 786648006656, 6823763935232, -15171034030080, -43440624908288, 92885896276480, 62882730299136, -195294436805376, 4878741111552, 168256221920320, -47884603550112, -74538290301632, 32698257976552, 18499228115810, -10836712986085, -2521472241195, 2084163747748, 142454592564, -243865756432, 7046247776, 17266725972, -1616621784, -707709554, 100635650, 15562316, -2923572, -173944, 43784, 852, -330, -1, 1]);
 

\( x^{31} - x^{30} - 330 x^{29} + 852 x^{28} + 43784 x^{27} - 173944 x^{26} - 2923572 x^{25} + 15562316 x^{24} + 100635650 x^{23} - 707709554 x^{22} - 1616621784 x^{21} + 17266725972 x^{20} + 7046247776 x^{19} - 243865756432 x^{18} + 142454592564 x^{17} + 2084163747748 x^{16} - 2521472241195 x^{15} - 10836712986085 x^{14} + 18499228115810 x^{13} + 32698257976552 x^{12} - 74538290301632 x^{11} - 47884603550112 x^{10} + 168256221920320 x^{9} + 4878741111552 x^{8} - 195294436805376 x^{7} + 62882730299136 x^{6} + 92885896276480 x^{5} - 43440624908288 x^{4} - 15171034030080 x^{3} + 6823763935232 x^{2} + 786648006656 x + 5192548352 \)

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

Invariants

Degree:  $31$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[31, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(107\!\cdots\!849\)\(\medspace = 683^{30}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $553.33$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $683$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $31$
This field is Galois and abelian over $\Q$.
Conductor:  \(683\)
Dirichlet character group:    $\lbrace$$\chi_{683}(253,·)$, $\chi_{683}(1,·)$, $\chi_{683}(603,·)$, $\chi_{683}(3,·)$, $\chi_{683}(646,·)$, $\chi_{683}(391,·)$, $\chi_{683}(9,·)$, $\chi_{683}(138,·)$, $\chi_{683}(76,·)$, $\chi_{683}(81,·)$, $\chi_{683}(67,·)$, $\chi_{683}(201,·)$, $\chi_{683}(27,·)$, $\chi_{683}(414,·)$, $\chi_{683}(367,·)$, $\chi_{683}(418,·)$, $\chi_{683}(347,·)$, $\chi_{683}(228,·)$, $\chi_{683}(358,·)$, $\chi_{683}(104,·)$, $\chi_{683}(490,·)$, $\chi_{683}(46,·)$, $\chi_{683}(559,·)$, $\chi_{683}(243,·)$, $\chi_{683}(350,·)$, $\chi_{683}(311,·)$, $\chi_{683}(312,·)$, $\chi_{683}(250,·)$, $\chi_{683}(571,·)$, $\chi_{683}(572,·)$, $\chi_{683}(443,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{7} + \frac{3}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{128} a^{8} - \frac{1}{32} a^{7} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} + \frac{1}{128} a^{4} - \frac{1}{32} a^{3} - \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{128} a^{9} + \frac{1}{64} a^{7} + \frac{1}{128} a^{5} - \frac{1}{32} a^{3}$, $\frac{1}{256} a^{10} - \frac{1}{256} a^{9} + \frac{3}{128} a^{7} - \frac{3}{256} a^{6} + \frac{15}{256} a^{5} - \frac{3}{128} a^{4} + \frac{3}{64} a^{3} + \frac{1}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{512} a^{11} - \frac{1}{512} a^{9} - \frac{1}{256} a^{8} - \frac{13}{512} a^{7} - \frac{1}{128} a^{6} - \frac{23}{512} a^{5} - \frac{1}{256} a^{4} + \frac{1}{128} a^{3} + \frac{1}{64} a^{2} + \frac{1}{16} a$, $\frac{1}{1024} a^{12} + \frac{1}{1024} a^{10} - \frac{1}{256} a^{9} + \frac{3}{1024} a^{8} - \frac{3}{128} a^{7} + \frac{3}{1024} a^{6} - \frac{9}{256} a^{5} + \frac{1}{128} a^{4} - \frac{1}{64} a^{2} + \frac{1}{16} a$, $\frac{1}{1024} a^{13} - \frac{1}{1024} a^{11} + \frac{1}{1024} a^{9} - \frac{1}{256} a^{8} - \frac{11}{1024} a^{7} - \frac{1}{128} a^{6} - \frac{7}{512} a^{5} - \frac{1}{256} a^{4} - \frac{5}{128} a^{3} + \frac{1}{64} a^{2} + \frac{1}{16} a$, $\frac{1}{8192} a^{14} + \frac{3}{8192} a^{13} - \frac{3}{8192} a^{12} + \frac{7}{8192} a^{11} - \frac{5}{8192} a^{10} + \frac{1}{8192} a^{9} - \frac{1}{8192} a^{8} + \frac{173}{8192} a^{7} + \frac{19}{1024} a^{6} - \frac{33}{1024} a^{5} + \frac{23}{512} a^{4} - \frac{43}{512} a^{3} - \frac{1}{16} a^{2} - \frac{13}{32} a - \frac{1}{2}$, $\frac{1}{8192} a^{15} - \frac{1}{2048} a^{13} - \frac{1}{4096} a^{11} - \frac{7}{2048} a^{9} - \frac{1}{256} a^{8} + \frac{153}{8192} a^{7} - \frac{1}{128} a^{6} + \frac{39}{1024} a^{5} - \frac{1}{256} a^{4} - \frac{107}{512} a^{3} + \frac{1}{64} a^{2} - \frac{11}{32} a - \frac{1}{2}$, $\frac{1}{32768} a^{16} + \frac{1}{8192} a^{13} + \frac{1}{16384} a^{12} - \frac{3}{8192} a^{11} - \frac{1}{512} a^{10} - \frac{5}{8192} a^{9} + \frac{69}{32768} a^{8} + \frac{191}{8192} a^{7} - \frac{27}{4096} a^{6} + \frac{3}{1024} a^{5} - \frac{67}{2048} a^{4} - \frac{125}{512} a^{3} + \frac{5}{128} a^{2} - \frac{9}{32} a - \frac{1}{2}$, $\frac{1}{32768} a^{17} - \frac{5}{16384} a^{13} - \frac{7}{8192} a^{11} + \frac{1}{32768} a^{9} - \frac{1}{256} a^{8} - \frac{179}{8192} a^{7} - \frac{1}{128} a^{6} - \frac{93}{2048} a^{5} - \frac{1}{256} a^{4} + \frac{115}{512} a^{3} + \frac{1}{64} a^{2} + \frac{11}{32} a - \frac{1}{2}$, $\frac{1}{65536} a^{18} - \frac{1}{65536} a^{17} - \frac{1}{16384} a^{15} - \frac{1}{32768} a^{14} + \frac{9}{32768} a^{13} + \frac{3}{16384} a^{12} + \frac{15}{16384} a^{11} - \frac{103}{65536} a^{10} - \frac{105}{65536} a^{9} - \frac{5}{16384} a^{8} + \frac{7}{4096} a^{7} + \frac{83}{4096} a^{6} + \frac{243}{4096} a^{5} + \frac{61}{1024} a^{4} - \frac{111}{512} a^{3} - \frac{5}{64} a^{2} - \frac{11}{32} a - \frac{1}{2}$, $\frac{1}{131072} a^{19} + \frac{1}{131072} a^{17} - \frac{1}{65536} a^{16} - \frac{3}{65536} a^{15} - \frac{1}{16384} a^{14} + \frac{5}{65536} a^{13} - \frac{11}{32768} a^{12} + \frac{29}{131072} a^{11} + \frac{13}{16384} a^{10} - \frac{251}{131072} a^{9} - \frac{33}{65536} a^{8} + \frac{177}{16384} a^{7} + \frac{227}{8192} a^{6} - \frac{11}{8192} a^{5} - \frac{161}{4096} a^{4} - \frac{9}{128} a^{3} + \frac{3}{256} a^{2} + \frac{1}{16} a$, $\frac{1}{262144} a^{20} - \frac{1}{262144} a^{18} - \frac{1}{131072} a^{16} + \frac{7}{131072} a^{14} + \frac{1}{4096} a^{13} - \frac{115}{262144} a^{12} - \frac{3}{4096} a^{11} - \frac{429}{262144} a^{10} - \frac{5}{4096} a^{9} + \frac{209}{65536} a^{8} - \frac{33}{4096} a^{7} - \frac{319}{16384} a^{6} + \frac{11}{512} a^{5} + \frac{155}{4096} a^{4} - \frac{43}{256} a^{3} - \frac{5}{256} a^{2} - \frac{11}{32} a - \frac{1}{2}$, $\frac{1}{1048576} a^{21} - \frac{1}{1048576} a^{20} - \frac{1}{1048576} a^{19} + \frac{1}{1048576} a^{18} - \frac{5}{524288} a^{17} + \frac{5}{524288} a^{16} + \frac{7}{524288} a^{15} - \frac{23}{524288} a^{14} + \frac{157}{1048576} a^{13} + \frac{227}{1048576} a^{12} - \frac{781}{1048576} a^{11} + \frac{77}{1048576} a^{10} - \frac{289}{262144} a^{9} + \frac{961}{262144} a^{8} + \frac{539}{65536} a^{7} + \frac{1443}{65536} a^{6} + \frac{77}{16384} a^{5} - \frac{153}{16384} a^{4} - \frac{343}{2048} a^{3} + \frac{239}{1024} a^{2} + \frac{5}{32} a - \frac{1}{4}$, $\frac{1}{2097152} a^{22} - \frac{1}{1048576} a^{20} - \frac{1}{262144} a^{19} + \frac{7}{2097152} a^{18} - \frac{3}{262144} a^{17} + \frac{1}{262144} a^{16} + \frac{5}{131072} a^{15} - \frac{113}{2097152} a^{14} - \frac{51}{131072} a^{13} - \frac{165}{1048576} a^{12} + \frac{51}{262144} a^{11} + \frac{2841}{2097152} a^{10} + \frac{745}{262144} a^{9} - \frac{135}{524288} a^{8} + \frac{373}{65536} a^{7} - \frac{2721}{131072} a^{6} + \frac{715}{16384} a^{5} - \frac{805}{32768} a^{4} - \frac{981}{4096} a^{3} - \frac{165}{2048} a^{2} - \frac{5}{16} a - \frac{3}{8}$, $\frac{1}{8388608} a^{23} - \frac{1}{4194304} a^{22} + \frac{1}{4194304} a^{21} - \frac{1}{1048576} a^{20} + \frac{19}{8388608} a^{19} - \frac{1}{4194304} a^{18} + \frac{3}{524288} a^{17} + \frac{5}{1048576} a^{16} - \frac{345}{8388608} a^{15} + \frac{221}{4194304} a^{14} - \frac{1371}{4194304} a^{13} - \frac{457}{1048576} a^{12} - \frac{7627}{8388608} a^{11} + \frac{565}{4194304} a^{10} + \frac{3737}{2097152} a^{9} - \frac{3131}{1048576} a^{8} + \frac{11207}{524288} a^{7} + \frac{4595}{262144} a^{6} - \frac{4565}{131072} a^{5} - \frac{3161}{65536} a^{4} - \frac{587}{4096} a^{3} - \frac{117}{4096} a^{2} - \frac{3}{32} a - \frac{3}{16}$, $\frac{1}{8388608} a^{24} - \frac{1}{4194304} a^{22} - \frac{1}{2097152} a^{21} + \frac{3}{8388608} a^{20} - \frac{7}{2097152} a^{19} + \frac{11}{2097152} a^{18} + \frac{9}{1048576} a^{17} + \frac{119}{8388608} a^{16} + \frac{17}{1048576} a^{15} - \frac{161}{4194304} a^{14} + \frac{627}{2097152} a^{13} + \frac{1701}{8388608} a^{12} - \frac{923}{2097152} a^{11} + \frac{679}{1048576} a^{10} + \frac{539}{524288} a^{9} + \frac{41}{131072} a^{8} + \frac{3477}{131072} a^{7} + \frac{391}{65536} a^{6} + \frac{1493}{32768} a^{5} - \frac{281}{32768} a^{4} - \frac{811}{4096} a^{3} + \frac{259}{2048} a^{2} - \frac{1}{8} a - \frac{3}{8}$, $\frac{1}{67108864} a^{25} + \frac{1}{67108864} a^{24} - \frac{5}{33554432} a^{22} + \frac{19}{67108864} a^{21} - \frac{57}{67108864} a^{20} - \frac{13}{33554432} a^{19} + \frac{959}{67108864} a^{17} - \frac{401}{67108864} a^{16} + \frac{317}{16777216} a^{15} + \frac{15}{33554432} a^{14} + \frac{22357}{67108864} a^{13} - \frac{21799}{67108864} a^{12} - \frac{1389}{33554432} a^{11} + \frac{20099}{16777216} a^{10} - \frac{23465}{8388608} a^{9} + \frac{12521}{4194304} a^{8} - \frac{12107}{2097152} a^{7} - \frac{32047}{1048576} a^{6} - \frac{23213}{524288} a^{5} - \frac{4835}{131072} a^{4} - \frac{6727}{32768} a^{3} + \frac{777}{8192} a^{2} + \frac{33}{128} a - \frac{1}{32}$, $\frac{1}{134217728} a^{26} - \frac{1}{134217728} a^{25} - \frac{1}{67108864} a^{24} + \frac{3}{67108864} a^{23} + \frac{7}{134217728} a^{22} + \frac{1}{134217728} a^{21} + \frac{19}{16777216} a^{20} - \frac{55}{33554432} a^{19} - \frac{289}{134217728} a^{18} - \frac{1679}{134217728} a^{17} - \frac{117}{67108864} a^{16} + \frac{19}{67108864} a^{15} - \frac{4807}{134217728} a^{14} + \frac{15375}{134217728} a^{13} + \frac{6701}{33554432} a^{12} + \frac{4701}{8388608} a^{11} - \frac{1905}{1048576} a^{10} + \frac{12867}{4194304} a^{9} + \frac{1797}{524288} a^{8} + \frac{3045}{262144} a^{7} + \frac{11811}{524288} a^{6} - \frac{16381}{524288} a^{5} + \frac{6399}{131072} a^{4} + \frac{2313}{32768} a^{3} + \frac{1705}{8192} a^{2} + \frac{25}{128} a - \frac{1}{32}$, $\frac{1}{536870912} a^{27} - \frac{1}{268435456} a^{26} + \frac{3}{536870912} a^{25} - \frac{5}{134217728} a^{24} - \frac{31}{536870912} a^{23} + \frac{9}{268435456} a^{22} - \frac{221}{536870912} a^{21} + \frac{25}{67108864} a^{20} - \frac{1933}{536870912} a^{19} - \frac{1847}{268435456} a^{18} + \frac{929}{536870912} a^{17} - \frac{1477}{134217728} a^{16} - \frac{3325}{536870912} a^{15} - \frac{5049}{268435456} a^{14} + \frac{41273}{536870912} a^{13} - \frac{2613}{16777216} a^{12} + \frac{35493}{67108864} a^{11} - \frac{17591}{33554432} a^{10} + \frac{2159}{2097152} a^{9} + \frac{21285}{8388608} a^{8} - \frac{52565}{4194304} a^{7} - \frac{1959}{2097152} a^{6} - \frac{57337}{2097152} a^{5} - \frac{29025}{524288} a^{4} + \frac{661}{131072} a^{3} - \frac{6151}{32768} a^{2} - \frac{111}{512} a - \frac{1}{128}$, $\frac{1}{8589934592} a^{28} + \frac{3}{8589934592} a^{27} + \frac{1}{8589934592} a^{26} + \frac{19}{8589934592} a^{25} + \frac{205}{8589934592} a^{24} + \frac{231}{8589934592} a^{23} - \frac{395}{8589934592} a^{22} - \frac{929}{8589934592} a^{21} - \frac{9797}{8589934592} a^{20} + \frac{11889}{8589934592} a^{19} - \frac{42509}{8589934592} a^{18} + \frac{113785}{8589934592} a^{17} + \frac{38127}{8589934592} a^{16} - \frac{392691}{8589934592} a^{15} - \frac{505209}{8589934592} a^{14} + \frac{1922581}{8589934592} a^{13} + \frac{256225}{1073741824} a^{12} + \frac{901275}{1073741824} a^{11} - \frac{843603}{536870912} a^{10} + \frac{307133}{134217728} a^{9} + \frac{144591}{134217728} a^{8} - \frac{1339703}{67108864} a^{7} + \frac{167835}{8388608} a^{6} - \frac{800489}{33554432} a^{5} + \frac{47599}{8388608} a^{4} + \frac{230085}{2097152} a^{3} + \frac{65805}{524288} a^{2} + \frac{1993}{8192} a - \frac{693}{2048}$, $\frac{1}{42537356099584} a^{29} - \frac{629}{21268678049792} a^{28} - \frac{9299}{21268678049792} a^{27} - \frac{62285}{21268678049792} a^{26} - \frac{118261}{21268678049792} a^{25} + \frac{631375}{21268678049792} a^{24} - \frac{107299}{21268678049792} a^{23} - \frac{2635289}{21268678049792} a^{22} - \frac{525387}{5317169512448} a^{21} + \frac{9286697}{21268678049792} a^{20} - \frac{32916005}{21268678049792} a^{19} - \frac{73417951}{21268678049792} a^{18} + \frac{5659717}{21268678049792} a^{17} - \frac{259682779}{21268678049792} a^{16} + \frac{308743407}{21268678049792} a^{15} + \frac{191040173}{21268678049792} a^{14} + \frac{18819551543}{42537356099584} a^{13} + \frac{1259650007}{2658584756224} a^{12} - \frac{2055089733}{5317169512448} a^{11} - \frac{2465265877}{2658584756224} a^{10} + \frac{1096215771}{332323094528} a^{9} - \frac{136404145}{664646189056} a^{8} - \frac{4140157469}{332323094528} a^{7} - \frac{2704993413}{166161547264} a^{6} + \frac{543512145}{166161547264} a^{5} + \frac{1436948785}{41540386816} a^{4} + \frac{2211314035}{10385096704} a^{3} - \frac{432606761}{2596274176} a^{2} + \frac{5071927}{40566784} a + \frac{2579}{16384}$, $\frac{1}{162998541765659601323562956891436190614782630716908840679157459786680725236684876152832} a^{30} + \frac{272551227831979449543217628572696890501278980824244197131106509221619419}{162998541765659601323562956891436190614782630716908840679157459786680725236684876152832} a^{29} + \frac{7436594867769738163234289946179405327690227991859139527060900223030788223}{159178263443026954417541950089293154897248662809481289725739706822930395738950074368} a^{28} + \frac{5282465763212194285511116034478336702232629868659332184584760810204767720301}{10187408860353725082722684805714761913423914419806802542447341236667545327292804759552} a^{27} - \frac{58293761590336684027428109978680749130015902074170460539049097364136772581741}{40749635441414900330890739222859047653695657679227210169789364946670181309171219038208} a^{26} - \frac{98532486190114621034448297816973054230129974807282940974405044357439481326835}{40749635441414900330890739222859047653695657679227210169789364946670181309171219038208} a^{25} + \frac{900725190135783097656146352310109999600611333196871737698090332671300053721045}{20374817720707450165445369611429523826847828839613605084894682473335090654585609519104} a^{24} - \frac{261841030429367088588946673173214221756388956952923648150090139319291788286177}{20374817720707450165445369611429523826847828839613605084894682473335090654585609519104} a^{23} - \frac{11345521095623882552303440878696424620147788055711472818919131063059547219791197}{81499270882829800661781478445718095307391315358454420339578729893340362618342438076416} a^{22} + \frac{14719548993268250662000063266954236323877412829036675124406413913909338265510129}{81499270882829800661781478445718095307391315358454420339578729893340362618342438076416} a^{21} - \frac{14164227520823661028781723612165573377540345101636456776345057430867602206806489}{20374817720707450165445369611429523826847828839613605084894682473335090654585609519104} a^{20} - \frac{35870680870658993939129743861144872595882407479212656343974060656237088317088655}{20374817720707450165445369611429523826847828839613605084894682473335090654585609519104} a^{19} - \frac{235621366930050961516515262105915423351643679035785297082774546436749671023457921}{40749635441414900330890739222859047653695657679227210169789364946670181309171219038208} a^{18} + \frac{168400189151164795014583943057748112168635294523946529218900559551333172060106013}{40749635441414900330890739222859047653695657679227210169789364946670181309171219038208} a^{17} - \frac{160088986263582834905672903573392428435393400076024762354345188280313295536341549}{20374817720707450165445369611429523826847828839613605084894682473335090654585609519104} a^{16} + \frac{657344114586458378949791346733595751871149856922487923826358028546036101513715185}{20374817720707450165445369611429523826847828839613605084894682473335090654585609519104} a^{15} - \frac{6543702924546329321435285094656743862726279331876442904457406609681281552721982847}{162998541765659601323562956891436190614782630716908840679157459786680725236684876152832} a^{14} - \frac{74462546361116686160249978802587816672460494261254607432587085834710811408668501541}{162998541765659601323562956891436190614782630716908840679157459786680725236684876152832} a^{13} + \frac{6713816352101540464388036551267664491905792010486018662323992464549319081094076377}{20374817720707450165445369611429523826847828839613605084894682473335090654585609519104} a^{12} + \frac{3795107980073848645918242848528932564902287981384129619967479504145933259828508069}{20374817720707450165445369611429523826847828839613605084894682473335090654585609519104} a^{11} + \frac{2658579194201255865758919471138194453563457937021541892768492843849301094989967943}{10187408860353725082722684805714761913423914419806802542447341236667545327292804759552} a^{10} - \frac{8193538171506941468703148934649870641670479994376019271475153521073582425036417867}{2546852215088431270680671201428690478355978604951700635611835309166886331823201189888} a^{9} - \frac{7476522934697031044996434395516373833963091406725659878224722030159823492150559399}{2546852215088431270680671201428690478355978604951700635611835309166886331823201189888} a^{8} - \frac{5002584972263978308928939208050516189841745125614299672754741610375905792856420835}{1273426107544215635340335600714345239177989302475850317805917654583443165911600594944} a^{7} + \frac{1283008053460547157591149486406600980724831869533708264812425705670034742075357715}{79589131721513477208770975044646577448624331404740644862869853411465197869475037184} a^{6} + \frac{15074537800010400195702493136008670954603724968438902134412299920750464025147441569}{636713053772107817670167800357172619588994651237925158902958827291721582955800297472} a^{5} - \frac{1221467167240029347773646175718473276538433384245336411301212812363713495166013367}{159178263443026954417541950089293154897248662809481289725739706822930395738950074368} a^{4} + \frac{7254504853722032217803573752708570063805218769613146935659663443268371344556909171}{39794565860756738604385487522323288724312165702370322431434926705732598934737518592} a^{3} - \frac{2397610147877722079075807852642111058098298971432419495837144795546139474551934581}{9948641465189184651096371880580822181078041425592580607858731676433149733684379648} a^{2} - \frac{66309271769394163938360851339998147515777230767163959809395505404667855111414129}{155447522893581010173380810634075346579344397274884071997792682444267964588818432} a - \frac{29130514358784409566732688970114377470528712532592158966854838749752373060521}{62781713608069874868085949367558702172594667720066264942565703733549258719232}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $30$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 588096262939769340000000000000000000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{31}\cdot(2\pi)^{0}\cdot 588096262939769340000000000000000000000000 \cdot 1}{2\sqrt{10780058321499447579019490584061143592974328542534512659762928215062178274195635731849}}\approx 1.92325882711620e8$ (assuming GRH)

Galois group

$C_{31}$ (as 31T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 31
The 31 conjugacy class representatives for $C_{31}$
Character table for $C_{31}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.1.0.1}{1} }^{31}$ $31$ $31$ $31$ $31$ $31$ $31$ $31$ $31$ $31$ $31$ $31$ $31$ $31$ $31$ $31$ $31$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
683Data not computed