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Properties

Label 40.0.56027829524...0625.1
Degree $40$
Signature $[0, 20]$
Discriminant $5^{20}\cdot 11^{36}\cdot 13^{20}$
Root discriminant $69.78$
Ramified primes $5, 11, 13$
Class number Not computed
Class group Not computed
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1048576, 0, 2359296, 0, 5046272, 0, 10764288, 0, 22958080, 0, 48964608, 0, 104430848, 0, 222728256, 0, 475030864, 0, 1013137380, 0, 2160801389, 0, 253284345, 0, 29689429, 0, 3480129, 0, 407933, 0, 47817, 0, 5605, 0, 657, 0, 77, 0, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 9*x^38 + 77*x^36 + 657*x^34 + 5605*x^32 + 47817*x^30 + 407933*x^28 + 3480129*x^26 + 29689429*x^24 + 253284345*x^22 + 2160801389*x^20 + 1013137380*x^18 + 475030864*x^16 + 222728256*x^14 + 104430848*x^12 + 48964608*x^10 + 22958080*x^8 + 10764288*x^6 + 5046272*x^4 + 2359296*x^2 + 1048576)
 
gp: K = bnfinit(x^40 + 9*x^38 + 77*x^36 + 657*x^34 + 5605*x^32 + 47817*x^30 + 407933*x^28 + 3480129*x^26 + 29689429*x^24 + 253284345*x^22 + 2160801389*x^20 + 1013137380*x^18 + 475030864*x^16 + 222728256*x^14 + 104430848*x^12 + 48964608*x^10 + 22958080*x^8 + 10764288*x^6 + 5046272*x^4 + 2359296*x^2 + 1048576, 1)
 

Normalized defining polynomial

\( x^{40} + 9 x^{38} + 77 x^{36} + 657 x^{34} + 5605 x^{32} + 47817 x^{30} + 407933 x^{28} + 3480129 x^{26} + 29689429 x^{24} + 253284345 x^{22} + 2160801389 x^{20} + 1013137380 x^{18} + 475030864 x^{16} + 222728256 x^{14} + 104430848 x^{12} + 48964608 x^{10} + 22958080 x^{8} + 10764288 x^{6} + 5046272 x^{4} + 2359296 x^{2} + 1048576 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(56027829524199487813298440814368728279159547663313566328064060211181640625=5^{20}\cdot 11^{36}\cdot 13^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.78$
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:  $5, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(715=5\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{715}(1,·)$, $\chi_{715}(259,·)$, $\chi_{715}(261,·)$, $\chi_{715}(519,·)$, $\chi_{715}(521,·)$, $\chi_{715}(651,·)$, $\chi_{715}(14,·)$, $\chi_{715}(144,·)$, $\chi_{715}(274,·)$, $\chi_{715}(131,·)$, $\chi_{715}(404,·)$, $\chi_{715}(534,·)$, $\chi_{715}(664,·)$, $\chi_{715}(324,·)$, $\chi_{715}(389,·)$, $\chi_{715}(326,·)$, $\chi_{715}(129,·)$, $\chi_{715}(391,·)$, $\chi_{715}(584,·)$, $\chi_{715}(51,·)$, $\chi_{715}(181,·)$, $\chi_{715}(311,·)$, $\chi_{715}(441,·)$, $\chi_{715}(571,·)$, $\chi_{715}(701,·)$, $\chi_{715}(64,·)$, $\chi_{715}(194,·)$, $\chi_{715}(196,·)$, $\chi_{715}(454,·)$, $\chi_{715}(456,·)$, $\chi_{715}(586,·)$, $\chi_{715}(79,·)$, $\chi_{715}(339,·)$, $\chi_{715}(469,·)$, $\chi_{715}(599,·)$, $\chi_{715}(714,·)$, $\chi_{715}(116,·)$, $\chi_{715}(246,·)$, $\chi_{715}(376,·)$, $\chi_{715}(636,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{10}$, $\frac{1}{8643205556} a^{22} - \frac{1}{4} a^{20} - \frac{1}{4} a^{18} - \frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{4} a^{12} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{253284345}{2160801389}$, $\frac{1}{17286411112} a^{23} + \frac{1}{8} a^{21} + \frac{1}{8} a^{19} + \frac{1}{8} a^{17} + \frac{1}{8} a^{15} + \frac{1}{8} a^{13} + \frac{1}{8} a^{11} - \frac{1}{2} a^{10} - \frac{3}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{2} a^{6} - \frac{3}{8} a^{5} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2} + \frac{253284345}{4321602778} a$, $\frac{1}{34572822224} a^{24} + \frac{1}{34572822224} a^{22} + \frac{1}{16} a^{20} - \frac{3}{16} a^{18} + \frac{1}{16} a^{16} - \frac{3}{16} a^{14} + \frac{1}{16} a^{12} - \frac{3}{16} a^{10} - \frac{1}{2} a^{9} + \frac{1}{16} a^{8} + \frac{5}{16} a^{6} - \frac{7}{16} a^{4} - \frac{1}{2} a^{3} - \frac{4068318433}{8643205556} a^{2} - \frac{1}{2} a - \frac{476879261}{2160801389}$, $\frac{1}{69145644448} a^{25} + \frac{1}{69145644448} a^{23} - \frac{7}{32} a^{21} - \frac{3}{32} a^{19} + \frac{1}{32} a^{17} + \frac{5}{32} a^{15} - \frac{7}{32} a^{13} - \frac{3}{32} a^{11} + \frac{1}{32} a^{9} - \frac{1}{2} a^{8} - \frac{11}{32} a^{7} - \frac{1}{2} a^{6} - \frac{7}{32} a^{5} - \frac{4068318433}{17286411112} a^{3} - \frac{1}{2} a^{2} - \frac{476879261}{4321602778} a$, $\frac{1}{138291288896} a^{26} + \frac{1}{138291288896} a^{24} + \frac{5}{138291288896} a^{22} + \frac{13}{64} a^{20} - \frac{15}{64} a^{18} + \frac{5}{64} a^{16} + \frac{9}{64} a^{14} - \frac{3}{64} a^{12} - \frac{31}{64} a^{10} - \frac{1}{2} a^{9} - \frac{11}{64} a^{8} + \frac{25}{64} a^{6} - \frac{4068318433}{34572822224} a^{4} - \frac{1}{2} a^{3} - \frac{476879261}{8643205556} a^{2} - \frac{1}{2} a - \frac{55898729}{2160801389}$, $\frac{1}{276582577792} a^{27} + \frac{1}{276582577792} a^{25} + \frac{5}{276582577792} a^{23} - \frac{19}{128} a^{21} - \frac{15}{128} a^{19} + \frac{5}{128} a^{17} - \frac{23}{128} a^{15} + \frac{29}{128} a^{13} - \frac{31}{128} a^{11} - \frac{1}{2} a^{10} - \frac{11}{128} a^{9} - \frac{1}{2} a^{8} + \frac{25}{128} a^{7} - \frac{4068318433}{69145644448} a^{5} - \frac{476879261}{17286411112} a^{3} - \frac{55898729}{4321602778} a$, $\frac{1}{553165155584} a^{28} + \frac{1}{553165155584} a^{26} + \frac{5}{553165155584} a^{24} - \frac{23}{553165155584} a^{22} - \frac{15}{256} a^{20} - \frac{59}{256} a^{18} + \frac{41}{256} a^{16} - \frac{35}{256} a^{14} + \frac{33}{256} a^{12} - \frac{75}{256} a^{10} - \frac{1}{2} a^{9} - \frac{39}{256} a^{8} - \frac{1}{2} a^{7} - \frac{38641140657}{138291288896} a^{6} - \frac{1}{2} a^{5} - \frac{9120084817}{34572822224} a^{4} - \frac{1108350059}{4321602778} a^{2} - \frac{259836670}{2160801389}$, $\frac{1}{1106330311168} a^{29} + \frac{1}{1106330311168} a^{27} + \frac{5}{1106330311168} a^{25} - \frac{23}{1106330311168} a^{23} + \frac{113}{512} a^{21} + \frac{69}{512} a^{19} - \frac{87}{512} a^{17} - \frac{35}{512} a^{15} + \frac{33}{512} a^{13} - \frac{75}{512} a^{11} - \frac{1}{2} a^{10} + \frac{217}{512} a^{9} - \frac{1}{2} a^{8} - \frac{38641140657}{276582577792} a^{7} + \frac{25452737407}{69145644448} a^{5} - \frac{1108350059}{8643205556} a^{3} - \frac{129918335}{2160801389} a$, $\frac{1}{4425321244672} a^{30} - \frac{1}{2212660622336} a^{29} + \frac{1}{4425321244672} a^{28} - \frac{1}{2212660622336} a^{27} + \frac{5}{4425321244672} a^{26} - \frac{5}{2212660622336} a^{25} + \frac{41}{4425321244672} a^{24} - \frac{41}{2212660622336} a^{23} - \frac{163}{4425321244672} a^{22} - \frac{177}{1024} a^{21} - \frac{123}{2048} a^{20} + \frac{123}{1024} a^{19} + \frac{233}{2048} a^{18} - \frac{233}{1024} a^{17} - \frac{483}{2048} a^{16} - \frac{29}{1024} a^{15} - \frac{159}{2048} a^{14} + \frac{159}{1024} a^{13} + \frac{501}{2048} a^{12} + \frac{11}{1024} a^{11} - \frac{999}{2048} a^{10} + \frac{487}{1024} a^{9} + \frac{272514259359}{1106330311168} a^{8} + \frac{4068318433}{553165155584} a^{7} + \frac{68668765187}{276582577792} a^{6} - \frac{68668765187}{138291288896} a^{5} - \frac{17342309841}{69145644448} a^{4} + \frac{55898729}{34572822224} a^{3} + \frac{4315050453}{17286411112} a^{2} + \frac{6552325}{8643205556} a - \frac{507336739}{4321602778}$, $\frac{1}{8850642489344} a^{31} - \frac{3}{8850642489344} a^{29} + \frac{1}{8850642489344} a^{27} + \frac{21}{8850642489344} a^{25} + \frac{185}{8850642489344} a^{23} - \frac{319}{4096} a^{21} - \frac{811}{4096} a^{19} + \frac{121}{4096} a^{17} + \frac{237}{4096} a^{15} - \frac{399}{4096} a^{13} - \frac{443}{4096} a^{11} - \frac{29044176579}{1106330311168} a^{9} - \frac{1}{2} a^{8} - \frac{68247784655}{138291288896} a^{7} - \frac{17181165979}{34572822224} a^{5} - \frac{4309266177}{8643205556} a^{3} - \frac{1}{2} a^{2} - \frac{2157909251}{4321602778} a - \frac{1}{2}$, $\frac{1}{17701284978688} a^{32} + \frac{1}{17701284978688} a^{30} - \frac{1}{2212660622336} a^{29} + \frac{5}{17701284978688} a^{28} - \frac{1}{2212660622336} a^{27} + \frac{41}{17701284978688} a^{26} - \frac{5}{2212660622336} a^{25} - \frac{163}{17701284978688} a^{24} - \frac{41}{2212660622336} a^{23} + \frac{417}{17701284978688} a^{22} + \frac{79}{1024} a^{21} - \frac{1815}{8192} a^{20} - \frac{133}{1024} a^{19} + \frac{541}{8192} a^{18} + \frac{23}{1024} a^{17} - \frac{159}{8192} a^{16} + \frac{227}{1024} a^{15} + \frac{501}{8192} a^{14} - \frac{97}{1024} a^{13} + \frac{1049}{8192} a^{12} - \frac{245}{1024} a^{11} + \frac{1932009726111}{4425321244672} a^{10} - \frac{281}{1024} a^{9} + \frac{206960054083}{1106330311168} a^{8} + \frac{142359607329}{553165155584} a^{7} + \frac{120948979055}{276582577792} a^{6} - \frac{34095942963}{138291288896} a^{5} - \frac{21614566215}{69145644448} a^{4} + \frac{8699104285}{34572822224} a^{3} - \frac{4828939517}{17286411112} a^{2} - \frac{538562266}{2160801389} a + \frac{447099803}{4321602778}$, $\frac{1}{35402569957376} a^{33} + \frac{1}{35402569957376} a^{31} - \frac{11}{35402569957376} a^{29} + \frac{25}{35402569957376} a^{27} - \frac{243}{35402569957376} a^{25} - \frac{239}{35402569957376} a^{23} - \frac{1}{17286411112} a^{22} - \frac{551}{16384} a^{21} - \frac{1}{8} a^{20} - \frac{1587}{16384} a^{19} - \frac{1}{8} a^{18} - \frac{3887}{16384} a^{17} - \frac{1}{8} a^{16} - \frac{4059}{16384} a^{15} - \frac{1}{8} a^{14} + \frac{3593}{16384} a^{13} - \frac{1}{8} a^{12} - \frac{185575635109}{8850642489344} a^{11} + \frac{3}{8} a^{10} + \frac{353052587471}{1106330311168} a^{9} + \frac{3}{8} a^{8} + \frac{7813581093}{34572822224} a^{7} + \frac{3}{8} a^{6} + \frac{6717567635}{69145644448} a^{5} + \frac{3}{8} a^{4} + \frac{241885298}{2160801389} a^{3} + \frac{3}{8} a^{2} - \frac{1933975325}{4321602778} a - \frac{253284345}{4321602778}$, $\frac{1}{70805139914752} a^{34} + \frac{1}{70805139914752} a^{32} + \frac{5}{70805139914752} a^{30} - \frac{1}{2212660622336} a^{29} + \frac{41}{70805139914752} a^{28} - \frac{1}{2212660622336} a^{27} - \frac{163}{70805139914752} a^{26} - \frac{5}{2212660622336} a^{25} + \frac{417}{70805139914752} a^{24} + \frac{23}{2212660622336} a^{23} - \frac{3787}{70805139914752} a^{22} + \frac{143}{1024} a^{21} - \frac{3555}{32768} a^{20} + \frac{187}{1024} a^{19} + \frac{8033}{32768} a^{18} + \frac{87}{1024} a^{17} + \frac{4597}{32768} a^{16} + \frac{35}{1024} a^{15} - \frac{7143}{32768} a^{14} - \frac{33}{1024} a^{13} + \frac{4144670348447}{17701284978688} a^{12} + \frac{75}{1024} a^{11} + \frac{1866455520835}{4425321244672} a^{10} + \frac{295}{1024} a^{9} - \frac{155633598737}{1106330311168} a^{8} - \frac{237941437135}{553165155584} a^{7} + \frac{12958256009}{276582577792} a^{6} + \frac{43692907041}{138291288896} a^{5} + \frac{21100677151}{69145644448} a^{4} + \frac{1108350059}{17286411112} a^{3} + \frac{4768702581}{17286411112} a^{2} + \frac{129918335}{4321602778} a - \frac{454160643}{4321602778}$, $\frac{1}{141610279829504} a^{35} + \frac{1}{141610279829504} a^{33} + \frac{5}{141610279829504} a^{31} - \frac{23}{141610279829504} a^{29} - \frac{227}{141610279829504} a^{27} + \frac{97}{141610279829504} a^{25} - \frac{1}{69145644448} a^{24} + \frac{1781}{141610279829504} a^{23} - \frac{1}{69145644448} a^{22} + \frac{9693}{65536} a^{21} + \frac{7}{32} a^{20} - \frac{8671}{65536} a^{19} + \frac{3}{32} a^{18} + \frac{14261}{65536} a^{17} - \frac{1}{32} a^{16} - \frac{807}{65536} a^{15} - \frac{5}{32} a^{14} - \frac{3634214651953}{35402569957376} a^{13} + \frac{7}{32} a^{12} + \frac{855200470783}{8850642489344} a^{11} - \frac{13}{32} a^{10} + \frac{310047049957}{1106330311168} a^{9} + \frac{15}{32} a^{8} + \frac{112231753893}{276582577792} a^{7} - \frac{5}{32} a^{6} - \frac{16213624787}{34572822224} a^{5} + \frac{7}{32} a^{4} + \frac{125749633}{8643205556} a^{3} + \frac{4068318433}{17286411112} a^{2} - \frac{2131321203}{4321602778} a + \frac{476879261}{4321602778}$, $\frac{1}{283220559659008} a^{36} + \frac{1}{283220559659008} a^{34} + \frac{5}{283220559659008} a^{32} - \frac{23}{283220559659008} a^{30} - \frac{227}{283220559659008} a^{28} + \frac{97}{283220559659008} a^{26} - \frac{1}{138291288896} a^{25} + \frac{1781}{283220559659008} a^{24} - \frac{1}{138291288896} a^{23} + \frac{15641}{283220559659008} a^{22} + \frac{7}{64} a^{21} + \frac{24097}{131072} a^{20} - \frac{13}{64} a^{19} - \frac{18507}{131072} a^{18} + \frac{15}{64} a^{17} - \frac{807}{131072} a^{16} - \frac{5}{64} a^{15} - \frac{3634214651953}{70805139914752} a^{14} - \frac{9}{64} a^{13} + \frac{855200470783}{17701284978688} a^{12} + \frac{3}{64} a^{11} - \frac{796283261211}{2212660622336} a^{10} - \frac{1}{64} a^{9} - \frac{26059535003}{553165155584} a^{8} + \frac{11}{64} a^{7} - \frac{33500035899}{69145644448} a^{6} - \frac{25}{64} a^{5} - \frac{8517455923}{17286411112} a^{4} + \frac{4068318433}{34572822224} a^{3} + \frac{14740093}{4321602778} a^{2} + \frac{476879261}{8643205556} a + \frac{3455602}{2160801389}$, $\frac{1}{566441119318016} a^{37} + \frac{1}{566441119318016} a^{35} + \frac{5}{566441119318016} a^{33} - \frac{23}{566441119318016} a^{31} - \frac{227}{566441119318016} a^{29} + \frac{97}{566441119318016} a^{27} - \frac{1}{276582577792} a^{26} + \frac{1781}{566441119318016} a^{25} - \frac{1}{276582577792} a^{24} + \frac{15641}{566441119318016} a^{23} - \frac{5}{276582577792} a^{22} + \frac{24097}{262144} a^{21} + \frac{19}{128} a^{20} + \frac{47029}{262144} a^{19} + \frac{15}{128} a^{18} + \frac{64729}{262144} a^{17} - \frac{5}{128} a^{16} + \frac{31768355305423}{141610279829504} a^{15} + \frac{23}{128} a^{14} - \frac{7995442018561}{35402569957376} a^{13} - \frac{29}{128} a^{12} + \frac{310047049957}{4425321244672} a^{11} - \frac{33}{128} a^{10} + \frac{527105620581}{1106330311168} a^{9} - \frac{53}{128} a^{8} + \frac{1072786325}{138291288896} a^{7} + \frac{39}{128} a^{6} - \frac{17160661479}{34572822224} a^{5} + \frac{4068318433}{69145644448} a^{4} - \frac{4306862685}{8643205556} a^{3} - \frac{8166326295}{17286411112} a^{2} - \frac{2157345787}{4321602778} a - \frac{1052451330}{2160801389}$, $\frac{1}{1132882238636032} a^{38} + \frac{1}{1132882238636032} a^{36} + \frac{5}{1132882238636032} a^{34} - \frac{23}{1132882238636032} a^{32} + \frac{29}{1132882238636032} a^{30} - \frac{1}{2212660622336} a^{29} + \frac{353}{1132882238636032} a^{28} + \frac{3}{2212660622336} a^{27} + \frac{3061}{1132882238636032} a^{26} - \frac{1}{2212660622336} a^{25} - \frac{6631}{1132882238636032} a^{24} - \frac{21}{2212660622336} a^{23} + \frac{59149}{1132882238636032} a^{22} - \frac{253}{1024} a^{21} - \frac{17227}{524288} a^{20} + \frac{63}{1024} a^{19} - \frac{39463}{524288} a^{18} + \frac{43}{1024} a^{17} + \frac{18077517704719}{283220559659008} a^{16} + \frac{135}{1024} a^{15} - \frac{216557018161}{70805139914752} a^{14} + \frac{19}{1024} a^{13} + \frac{1922004886151}{8850642489344} a^{12} + \frac{143}{1024} a^{11} - \frac{274681613073}{4425321244672} a^{10} + \frac{187}{1024} a^{9} - \frac{345505395373}{1106330311168} a^{8} + \frac{98189821027}{276582577792} a^{7} - \frac{17260291841}{276582577792} a^{6} + \frac{7745345763}{34572822224} a^{5} - \frac{21604952247}{69145644448} a^{4} + \frac{513889064}{2160801389} a^{3} - \frac{4827812589}{17286411112} a^{2} - \frac{2210147793}{8643205556} a + \frac{447231899}{4321602778}$, $\frac{1}{2265764477272064} a^{39} + \frac{1}{2265764477272064} a^{37} + \frac{5}{2265764477272064} a^{35} - \frac{23}{2265764477272064} a^{33} + \frac{29}{2265764477272064} a^{31} - \frac{671}{2265764477272064} a^{29} - \frac{1}{1106330311168} a^{28} + \frac{2037}{2265764477272064} a^{27} - \frac{1}{1106330311168} a^{26} - \frac{11751}{2265764477272064} a^{25} - \frac{5}{1106330311168} a^{24} + \frac{17165}{2265764477272064} a^{23} + \frac{23}{1106330311168} a^{22} - \frac{198475}{1048576} a^{21} - \frac{113}{512} a^{20} - \frac{175655}{1048576} a^{19} - \frac{69}{512} a^{18} + \frac{30800316283151}{566441119318016} a^{17} + \frac{87}{512} a^{16} + \frac{31175565561231}{141610279829504} a^{15} + \frac{35}{512} a^{14} - \frac{4180098236385}{17701284978688} a^{13} - \frac{33}{512} a^{12} - \frac{179606351957}{8850642489344} a^{11} - \frac{181}{512} a^{10} - \frac{199762715049}{1106330311168} a^{9} - \frac{217}{512} a^{8} + \frac{3909353609}{17286411112} a^{7} - \frac{99650148239}{276582577792} a^{6} + \frac{6722374619}{69145644448} a^{5} - \frac{25452737407}{69145644448} a^{4} - \frac{1676889927}{4321602778} a^{3} + \frac{1108350059}{8643205556} a^{2} - \frac{1933909277}{4321602778} a - \frac{1900964719}{4321602778}$

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:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1}{8643205556} a^{24} - \frac{18434075121}{8643205556} a^{2} \) (order $22$)
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

$C_2^2\times C_{10}$ (as 40T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-715}) \), \(\Q(\sqrt{-143}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{13}, \sqrt{-55})\), \(\Q(\sqrt{-11}, \sqrt{13})\), \(\Q(\sqrt{-55}, \sqrt{65})\), \(\Q(\sqrt{-11}, \sqrt{65})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{5}, \sqrt{-143})\), \(\Q(\zeta_{11})^+\), 8.0.261351000625.1, 10.10.79589952003133.1, 10.10.248718600009790625.1, 10.10.669871503125.1, 10.0.7368586534375.1, 10.0.2735904600107696875.1, 10.0.875489472034463.1, \(\Q(\zeta_{11})\), 20.20.61860941990830221086345856337890625.1, 20.0.7485173980890456751447848616884765625.2, 20.0.766481815643182771348259698369.1, 20.0.7485173980890456751447848616884765625.1, 20.0.7485173980890456751447848616884765625.4, 20.0.54296067514572573056640625.1, 20.0.7485173980890456751447848616884765625.3

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ R R ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
5Data not computed
11Data not computed
13Data not computed