Properties

Label 40.0.560...625.1
Degree $40$
Signature $[0, 20]$
Discriminant $5.603\times 10^{73}$
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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Normalized defining polynomial

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(y^40 + 9*y^38 + 77*y^36 + 657*y^34 + 5605*y^32 + 47817*y^30 + 407933*y^28 + 3480129*y^26 + 29689429*y^24 + 253284345*y^22 + 2160801389*y^20 + 1013137380*y^18 + 475030864*y^16 + 222728256*y^14 + 104430848*y^12 + 48964608*y^10 + 22958080*y^8 + 10764288*y^6 + 5046272*y^4 + 2359296*y^2 + 1048576, 1)
 
magma: R<x> := PolynomialRing(Rationals()); 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);
 
oscar: Qx, x = PolynomialRing(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)
 

\( x^{40} + 9 x^{38} + 77 x^{36} + 657 x^{34} + 5605 x^{32} + 47817 x^{30} + 407933 x^{28} + 3480129 x^{26} + 29689429 x^{24} + 253284345 x^{22} + 2160801389 x^{20} + \cdots + 1048576 \) Copy content Toggle raw display

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

Invariants

Degree:  $40$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 20]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(56027829524199487813298440814368728279159547663313566328064060211181640625\) \(\medspace = 5^{20}\cdot 11^{36}\cdot 13^{20}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(69.78\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $5^{1/2}11^{9/10}13^{1/2}\approx 69.77664678227555$
Ramified primes:   \(5\), \(11\), \(13\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $40$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
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.
Reflex fields:  unavailable$^{524288}$

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

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

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $19$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{1}{8643205556} a^{24} - \frac{18434075121}{8643205556} a^{2} \)  (order $22$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
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)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
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);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(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);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(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);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
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.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.10.0.1}{10} }^{4}$ ${\href{/padicField/3.10.0.1}{10} }^{4}$ R ${\href{/padicField/7.10.0.1}{10} }^{4}$ R R ${\href{/padicField/17.10.0.1}{10} }^{4}$ ${\href{/padicField/19.10.0.1}{10} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{20}$ ${\href{/padicField/29.10.0.1}{10} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{4}$ ${\href{/padicField/37.10.0.1}{10} }^{4}$ ${\href{/padicField/41.10.0.1}{10} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{20}$ ${\href{/padicField/47.10.0.1}{10} }^{4}$ ${\href{/padicField/53.10.0.1}{10} }^{4}$ ${\href{/padicField/59.10.0.1}{10} }^{4}$

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

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display Deg $20$$2$$10$$10$
Deg $20$$2$$10$$10$
\(11\) Copy content Toggle raw display 11.20.18.1$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$$10$$2$$18$20T3$[\ ]_{10}^{2}$
11.20.18.1$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$$10$$2$$18$20T3$[\ ]_{10}^{2}$
\(13\) Copy content Toggle raw display 13.20.10.1$x^{20} + 1300 x^{19} + 760630 x^{18} + 263792100 x^{17} + 60057199605 x^{16} + 9380582749214 x^{15} + 1018311780763300 x^{14} + 75907334238787106 x^{13} + 3723649138838156452 x^{12} + 108997457660124507008 x^{11} + 1475055936539742904023 x^{10} + 1416974086834508372240 x^{9} + 629829636823684464902 x^{8} + 192953738203144224036 x^{7} + 798096976259371806456 x^{6} + 10514315760388324731550 x^{5} + 12773049284749524150248 x^{4} + 15044084853570106847092 x^{3} + 5322693419045842540420 x^{2} + 2225999864943544940488 x + 2898376971411614164801$$2$$10$$10$20T3$[\ ]_{2}^{10}$
13.20.10.1$x^{20} + 1300 x^{19} + 760630 x^{18} + 263792100 x^{17} + 60057199605 x^{16} + 9380582749214 x^{15} + 1018311780763300 x^{14} + 75907334238787106 x^{13} + 3723649138838156452 x^{12} + 108997457660124507008 x^{11} + 1475055936539742904023 x^{10} + 1416974086834508372240 x^{9} + 629829636823684464902 x^{8} + 192953738203144224036 x^{7} + 798096976259371806456 x^{6} + 10514315760388324731550 x^{5} + 12773049284749524150248 x^{4} + 15044084853570106847092 x^{3} + 5322693419045842540420 x^{2} + 2225999864943544940488 x + 2898376971411614164801$$2$$10$$10$20T3$[\ ]_{2}^{10}$