Properties

Label 42.0.409...667.1
Degree $42$
Signature $[0, 21]$
Discriminant $-4.092\times 10^{86}$
Root discriminant \(115.40\)
Ramified primes $7,29$
Class number not computed
Class group not computed
Galois group $C_{42}$ (as 42T1)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 - x^41 + 20*x^40 - 25*x^39 - 8*x^38 - 138*x^37 - 2077*x^36 + 839*x^35 - 7305*x^34 + 8805*x^33 + 67596*x^32 - 12107*x^31 + 444232*x^30 - 141646*x^29 + 238560*x^28 + 2150502*x^27 - 3285732*x^26 + 14361861*x^25 - 708679*x^24 + 4004385*x^23 + 63920534*x^22 - 73366557*x^21 + 232544983*x^20 + 114300646*x^19 + 134380032*x^18 + 644451241*x^17 - 110366019*x^16 + 50739485*x^15 + 3000292231*x^14 - 3874326960*x^13 + 7531027885*x^12 - 11845394651*x^11 + 8115584321*x^10 - 10398559445*x^9 + 11442787794*x^8 - 3901741530*x^7 + 3575301015*x^6 - 3614499453*x^5 - 715331685*x^4 + 331663193*x^3 + 2507361126*x^2 - 1452650295*x + 649728353)
 
gp: K = bnfinit(y^42 - y^41 + 20*y^40 - 25*y^39 - 8*y^38 - 138*y^37 - 2077*y^36 + 839*y^35 - 7305*y^34 + 8805*y^33 + 67596*y^32 - 12107*y^31 + 444232*y^30 - 141646*y^29 + 238560*y^28 + 2150502*y^27 - 3285732*y^26 + 14361861*y^25 - 708679*y^24 + 4004385*y^23 + 63920534*y^22 - 73366557*y^21 + 232544983*y^20 + 114300646*y^19 + 134380032*y^18 + 644451241*y^17 - 110366019*y^16 + 50739485*y^15 + 3000292231*y^14 - 3874326960*y^13 + 7531027885*y^12 - 11845394651*y^11 + 8115584321*y^10 - 10398559445*y^9 + 11442787794*y^8 - 3901741530*y^7 + 3575301015*y^6 - 3614499453*y^5 - 715331685*y^4 + 331663193*y^3 + 2507361126*y^2 - 1452650295*y + 649728353, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^42 - x^41 + 20*x^40 - 25*x^39 - 8*x^38 - 138*x^37 - 2077*x^36 + 839*x^35 - 7305*x^34 + 8805*x^33 + 67596*x^32 - 12107*x^31 + 444232*x^30 - 141646*x^29 + 238560*x^28 + 2150502*x^27 - 3285732*x^26 + 14361861*x^25 - 708679*x^24 + 4004385*x^23 + 63920534*x^22 - 73366557*x^21 + 232544983*x^20 + 114300646*x^19 + 134380032*x^18 + 644451241*x^17 - 110366019*x^16 + 50739485*x^15 + 3000292231*x^14 - 3874326960*x^13 + 7531027885*x^12 - 11845394651*x^11 + 8115584321*x^10 - 10398559445*x^9 + 11442787794*x^8 - 3901741530*x^7 + 3575301015*x^6 - 3614499453*x^5 - 715331685*x^4 + 331663193*x^3 + 2507361126*x^2 - 1452650295*x + 649728353);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^42 - x^41 + 20*x^40 - 25*x^39 - 8*x^38 - 138*x^37 - 2077*x^36 + 839*x^35 - 7305*x^34 + 8805*x^33 + 67596*x^32 - 12107*x^31 + 444232*x^30 - 141646*x^29 + 238560*x^28 + 2150502*x^27 - 3285732*x^26 + 14361861*x^25 - 708679*x^24 + 4004385*x^23 + 63920534*x^22 - 73366557*x^21 + 232544983*x^20 + 114300646*x^19 + 134380032*x^18 + 644451241*x^17 - 110366019*x^16 + 50739485*x^15 + 3000292231*x^14 - 3874326960*x^13 + 7531027885*x^12 - 11845394651*x^11 + 8115584321*x^10 - 10398559445*x^9 + 11442787794*x^8 - 3901741530*x^7 + 3575301015*x^6 - 3614499453*x^5 - 715331685*x^4 + 331663193*x^3 + 2507361126*x^2 - 1452650295*x + 649728353)
 

\( x^{42} - x^{41} + 20 x^{40} - 25 x^{39} - 8 x^{38} - 138 x^{37} - 2077 x^{36} + 839 x^{35} - 7305 x^{34} + 8805 x^{33} + 67596 x^{32} - 12107 x^{31} + 444232 x^{30} + \cdots + 649728353 \) Copy content Toggle raw display

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 21]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-409\!\cdots\!667\) \(\medspace = -\,7^{35}\cdot 29^{39}\) 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:  \(115.40\)
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:  $7^{5/6}29^{13/14}\approx 115.39563058619058$
Ramified primes:   \(7\), \(29\) 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:  \(\Q(\sqrt{-203}) \)
$\card{ \Gal(K/\Q) }$:  $42$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(203=7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{203}(1,·)$, $\chi_{203}(5,·)$, $\chi_{203}(6,·)$, $\chi_{203}(129,·)$, $\chi_{203}(138,·)$, $\chi_{203}(13,·)$, $\chi_{203}(16,·)$, $\chi_{203}(150,·)$, $\chi_{203}(23,·)$, $\chi_{203}(25,·)$, $\chi_{203}(30,·)$, $\chi_{203}(33,·)$, $\chi_{203}(34,·)$, $\chi_{203}(36,·)$, $\chi_{203}(165,·)$, $\chi_{203}(38,·)$, $\chi_{203}(167,·)$, $\chi_{203}(169,·)$, $\chi_{203}(170,·)$, $\chi_{203}(173,·)$, $\chi_{203}(178,·)$, $\chi_{203}(180,·)$, $\chi_{203}(53,·)$, $\chi_{203}(187,·)$, $\chi_{203}(74,·)$, $\chi_{203}(62,·)$, $\chi_{203}(65,·)$, $\chi_{203}(197,·)$, $\chi_{203}(198,·)$, $\chi_{203}(202,·)$, $\chi_{203}(78,·)$, $\chi_{203}(141,·)$, $\chi_{203}(80,·)$, $\chi_{203}(81,·)$, $\chi_{203}(88,·)$, $\chi_{203}(96,·)$, $\chi_{203}(107,·)$, $\chi_{203}(115,·)$, $\chi_{203}(190,·)$, $\chi_{203}(122,·)$, $\chi_{203}(123,·)$, $\chi_{203}(125,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{1048576}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{41}a^{25}+\frac{11}{41}a^{24}+\frac{16}{41}a^{23}-\frac{8}{41}a^{22}+\frac{4}{41}a^{21}+\frac{15}{41}a^{20}+\frac{5}{41}a^{19}-\frac{2}{41}a^{18}+\frac{16}{41}a^{17}-\frac{16}{41}a^{16}+\frac{2}{41}a^{15}-\frac{11}{41}a^{14}-\frac{4}{41}a^{13}-\frac{8}{41}a^{12}-\frac{5}{41}a^{11}-\frac{1}{41}a^{10}+\frac{14}{41}a^{9}-\frac{14}{41}a^{8}+\frac{4}{41}a^{7}-\frac{7}{41}a^{6}-\frac{6}{41}a^{5}+\frac{15}{41}a^{4}-\frac{19}{41}a^{3}+\frac{18}{41}a^{2}+\frac{4}{41}a$, $\frac{1}{41}a^{26}+\frac{18}{41}a^{24}-\frac{20}{41}a^{23}+\frac{10}{41}a^{22}+\frac{12}{41}a^{21}+\frac{4}{41}a^{20}-\frac{16}{41}a^{19}-\frac{3}{41}a^{18}+\frac{13}{41}a^{17}+\frac{14}{41}a^{16}+\frac{8}{41}a^{15}-\frac{6}{41}a^{14}-\frac{5}{41}a^{13}+\frac{1}{41}a^{12}+\frac{13}{41}a^{11}-\frac{16}{41}a^{10}-\frac{4}{41}a^{9}-\frac{6}{41}a^{8}-\frac{10}{41}a^{7}-\frac{11}{41}a^{6}-\frac{1}{41}a^{5}-\frac{20}{41}a^{4}-\frac{19}{41}a^{3}+\frac{11}{41}a^{2}-\frac{3}{41}a$, $\frac{1}{41}a^{27}-\frac{13}{41}a^{24}+\frac{9}{41}a^{23}-\frac{8}{41}a^{22}+\frac{14}{41}a^{21}+\frac{1}{41}a^{20}-\frac{11}{41}a^{19}+\frac{8}{41}a^{18}+\frac{13}{41}a^{17}+\frac{9}{41}a^{16}-\frac{1}{41}a^{15}-\frac{12}{41}a^{14}-\frac{9}{41}a^{13}-\frac{7}{41}a^{12}-\frac{8}{41}a^{11}+\frac{14}{41}a^{10}-\frac{12}{41}a^{9}-\frac{4}{41}a^{8}-\frac{1}{41}a^{7}+\frac{2}{41}a^{6}+\frac{6}{41}a^{5}-\frac{2}{41}a^{4}-\frac{16}{41}a^{3}+\frac{1}{41}a^{2}+\frac{10}{41}a$, $\frac{1}{41}a^{28}-\frac{12}{41}a^{24}-\frac{5}{41}a^{23}-\frac{8}{41}a^{22}+\frac{12}{41}a^{21}+\frac{20}{41}a^{20}-\frac{9}{41}a^{19}-\frac{13}{41}a^{18}+\frac{12}{41}a^{17}-\frac{4}{41}a^{16}+\frac{14}{41}a^{15}+\frac{12}{41}a^{14}-\frac{18}{41}a^{13}+\frac{11}{41}a^{12}-\frac{10}{41}a^{11}+\frac{16}{41}a^{10}+\frac{14}{41}a^{9}-\frac{19}{41}a^{8}+\frac{13}{41}a^{7}-\frac{3}{41}a^{6}+\frac{2}{41}a^{5}+\frac{15}{41}a^{4}-\frac{2}{41}a^{2}+\frac{11}{41}a$, $\frac{1}{41}a^{29}+\frac{4}{41}a^{24}+\frac{20}{41}a^{23}-\frac{2}{41}a^{22}-\frac{14}{41}a^{21}+\frac{7}{41}a^{20}+\frac{6}{41}a^{19}-\frac{12}{41}a^{18}-\frac{17}{41}a^{17}-\frac{14}{41}a^{16}-\frac{5}{41}a^{15}+\frac{14}{41}a^{14}+\frac{4}{41}a^{13}+\frac{17}{41}a^{12}-\frac{3}{41}a^{11}+\frac{2}{41}a^{10}-\frac{15}{41}a^{9}+\frac{9}{41}a^{8}+\frac{4}{41}a^{7}-\frac{16}{41}a^{5}+\frac{16}{41}a^{4}+\frac{16}{41}a^{3}-\frac{19}{41}a^{2}+\frac{7}{41}a$, $\frac{1}{41}a^{30}+\frac{17}{41}a^{24}+\frac{16}{41}a^{23}+\frac{18}{41}a^{22}-\frac{9}{41}a^{21}-\frac{13}{41}a^{20}+\frac{9}{41}a^{19}-\frac{9}{41}a^{18}+\frac{4}{41}a^{17}+\frac{18}{41}a^{16}+\frac{6}{41}a^{15}+\frac{7}{41}a^{14}-\frac{8}{41}a^{13}-\frac{12}{41}a^{12}-\frac{19}{41}a^{11}-\frac{11}{41}a^{10}-\frac{6}{41}a^{9}+\frac{19}{41}a^{8}-\frac{16}{41}a^{7}+\frac{12}{41}a^{6}-\frac{1}{41}a^{5}-\frac{3}{41}a^{4}+\frac{16}{41}a^{3}+\frac{17}{41}a^{2}-\frac{16}{41}a$, $\frac{1}{41}a^{31}-\frac{7}{41}a^{24}-\frac{8}{41}a^{23}+\frac{4}{41}a^{22}+\frac{1}{41}a^{21}-\frac{12}{41}a^{19}-\frac{3}{41}a^{18}-\frac{8}{41}a^{17}-\frac{9}{41}a^{16}+\frac{14}{41}a^{15}+\frac{15}{41}a^{14}+\frac{15}{41}a^{13}-\frac{6}{41}a^{12}-\frac{8}{41}a^{11}+\frac{11}{41}a^{10}-\frac{14}{41}a^{9}+\frac{17}{41}a^{8}-\frac{15}{41}a^{7}-\frac{5}{41}a^{6}+\frac{17}{41}a^{5}+\frac{7}{41}a^{4}+\frac{12}{41}a^{3}+\frac{6}{41}a^{2}+\frac{14}{41}a$, $\frac{1}{41}a^{32}-\frac{13}{41}a^{24}-\frac{7}{41}a^{23}-\frac{14}{41}a^{22}-\frac{13}{41}a^{21}+\frac{11}{41}a^{20}-\frac{9}{41}a^{19}+\frac{19}{41}a^{18}-\frac{20}{41}a^{17}-\frac{16}{41}a^{16}-\frac{12}{41}a^{15}+\frac{20}{41}a^{14}+\frac{7}{41}a^{13}+\frac{18}{41}a^{12}+\frac{17}{41}a^{11}+\frac{20}{41}a^{10}-\frac{8}{41}a^{9}+\frac{10}{41}a^{8}-\frac{18}{41}a^{7}+\frac{9}{41}a^{6}+\frac{6}{41}a^{5}-\frac{6}{41}a^{4}-\frac{4}{41}a^{3}+\frac{17}{41}a^{2}-\frac{13}{41}a$, $\frac{1}{41}a^{33}+\frac{13}{41}a^{24}-\frac{11}{41}a^{23}+\frac{6}{41}a^{22}-\frac{19}{41}a^{21}-\frac{19}{41}a^{20}+\frac{2}{41}a^{19}-\frac{5}{41}a^{18}-\frac{13}{41}a^{17}-\frac{15}{41}a^{16}+\frac{5}{41}a^{15}-\frac{13}{41}a^{14}+\frac{7}{41}a^{13}-\frac{5}{41}a^{12}-\frac{4}{41}a^{11}+\frac{20}{41}a^{10}-\frac{13}{41}a^{9}+\frac{5}{41}a^{8}+\frac{20}{41}a^{7}-\frac{3}{41}a^{6}-\frac{2}{41}a^{5}-\frac{14}{41}a^{4}+\frac{16}{41}a^{3}+\frac{16}{41}a^{2}+\frac{11}{41}a$, $\frac{1}{41}a^{34}+\frac{10}{41}a^{24}+\frac{3}{41}a^{23}+\frac{3}{41}a^{22}+\frac{11}{41}a^{21}+\frac{12}{41}a^{20}+\frac{12}{41}a^{19}+\frac{13}{41}a^{18}-\frac{18}{41}a^{17}+\frac{8}{41}a^{16}+\frac{2}{41}a^{15}-\frac{14}{41}a^{14}+\frac{6}{41}a^{13}+\frac{18}{41}a^{12}+\frac{3}{41}a^{11}-\frac{13}{41}a^{9}-\frac{3}{41}a^{8}-\frac{14}{41}a^{7}+\frac{7}{41}a^{6}-\frac{18}{41}a^{5}-\frac{15}{41}a^{4}+\frac{17}{41}a^{3}-\frac{18}{41}a^{2}-\frac{11}{41}a$, $\frac{1}{1681}a^{35}+\frac{5}{1681}a^{34}-\frac{4}{1681}a^{33}-\frac{6}{1681}a^{32}-\frac{10}{1681}a^{31}-\frac{6}{1681}a^{30}+\frac{10}{1681}a^{29}-\frac{9}{1681}a^{28}-\frac{13}{1681}a^{27}-\frac{12}{1681}a^{26}+\frac{16}{1681}a^{25}-\frac{32}{1681}a^{24}+\frac{511}{1681}a^{23}-\frac{320}{1681}a^{22}-\frac{654}{1681}a^{21}+\frac{366}{1681}a^{20}-\frac{580}{1681}a^{19}-\frac{743}{1681}a^{18}+\frac{131}{1681}a^{17}+\frac{23}{1681}a^{16}+\frac{240}{1681}a^{15}-\frac{7}{1681}a^{14}+\frac{149}{1681}a^{13}+\frac{75}{1681}a^{12}+\frac{306}{1681}a^{11}+\frac{443}{1681}a^{10}+\frac{179}{1681}a^{9}-\frac{748}{1681}a^{8}-\frac{283}{1681}a^{7}-\frac{120}{1681}a^{6}-\frac{3}{1681}a^{5}+\frac{650}{1681}a^{4}+\frac{662}{1681}a^{3}-\frac{351}{1681}a^{2}+\frac{3}{41}a$, $\frac{1}{1681}a^{36}+\frac{12}{1681}a^{34}+\frac{14}{1681}a^{33}+\frac{20}{1681}a^{32}+\frac{3}{1681}a^{31}-\frac{1}{1681}a^{30}-\frac{18}{1681}a^{29}-\frac{9}{1681}a^{28}+\frac{12}{1681}a^{27}-\frac{6}{1681}a^{26}+\frac{11}{1681}a^{25}+\frac{56}{1681}a^{24}-\frac{497}{1681}a^{23}+\frac{618}{1681}a^{22}+\frac{602}{1681}a^{21}-\frac{442}{1681}a^{20}+\frac{722}{1681}a^{19}-\frac{459}{1681}a^{18}-\frac{345}{1681}a^{17}-\frac{449}{1681}a^{16}+\frac{269}{1681}a^{15}+\frac{102}{1681}a^{14}+\frac{478}{1681}a^{13}-\frac{807}{1681}a^{12}+\frac{758}{1681}a^{11}-\frac{314}{1681}a^{10}-\frac{3}{1681}a^{9}+\frac{259}{1681}a^{8}-\frac{386}{1681}a^{7}+\frac{679}{1681}a^{6}-\frac{688}{1681}a^{5}+\frac{241}{1681}a^{4}-\frac{217}{1681}a^{3}+\frac{771}{1681}a^{2}-\frac{20}{41}a$, $\frac{1}{1681}a^{37}-\frac{5}{1681}a^{34}-\frac{14}{1681}a^{33}-\frac{7}{1681}a^{32}-\frac{4}{1681}a^{31}+\frac{13}{1681}a^{30}-\frac{6}{1681}a^{29}-\frac{3}{1681}a^{28}-\frac{14}{1681}a^{27}-\frac{9}{1681}a^{26}-\frac{13}{1681}a^{25}-\frac{400}{1681}a^{24}-\frac{102}{1681}a^{23}+\frac{55}{1681}a^{22}+\frac{395}{1681}a^{21}+\frac{799}{1681}a^{20}+\frac{433}{1681}a^{19}-\frac{654}{1681}a^{18}-\frac{53}{1681}a^{17}-\frac{376}{1681}a^{16}-\frac{605}{1681}a^{15}+\frac{808}{1681}a^{14}-\frac{504}{1681}a^{13}-\frac{183}{1681}a^{12}-\frac{378}{1681}a^{11}+\frac{749}{1681}a^{10}+\frac{366}{1681}a^{9}-\frac{676}{1681}a^{8}+\frac{303}{1681}a^{7}-\frac{27}{1681}a^{6}+\frac{113}{1681}a^{5}-\frac{473}{1681}a^{4}+\frac{822}{1681}a^{3}+\frac{30}{1681}a^{2}-\frac{15}{41}a$, $\frac{1}{1681}a^{38}+\frac{11}{1681}a^{34}+\frac{14}{1681}a^{33}+\frac{7}{1681}a^{32}+\frac{4}{1681}a^{31}+\frac{5}{1681}a^{30}+\frac{6}{1681}a^{29}-\frac{18}{1681}a^{28}+\frac{8}{1681}a^{27}+\frac{9}{1681}a^{26}+\frac{8}{1681}a^{25}+\frac{148}{1681}a^{24}+\frac{478}{1681}a^{23}+\frac{25}{1681}a^{22}+\frac{399}{1681}a^{21}+\frac{541}{1681}a^{20}-\frac{110}{1681}a^{19}-\frac{611}{1681}a^{18}+\frac{607}{1681}a^{17}+\frac{699}{1681}a^{16}-\frac{493}{1681}a^{15}+\frac{527}{1681}a^{14}-\frac{258}{1681}a^{13}-\frac{208}{1681}a^{12}+\frac{188}{1681}a^{11}-\frac{740}{1681}a^{10}-\frac{355}{1681}a^{9}+\frac{499}{1681}a^{8}-\frac{171}{1681}a^{7}+\frac{251}{1681}a^{6}-\frac{488}{1681}a^{5}-\frac{233}{1681}a^{4}+\frac{265}{1681}a^{3}+\frac{787}{1681}a^{2}+\frac{20}{41}a$, $\frac{1}{17824555783}a^{39}-\frac{1886705}{17824555783}a^{38}+\frac{3048103}{17824555783}a^{37}+\frac{131785}{17824555783}a^{36}-\frac{2537640}{17824555783}a^{35}-\frac{121502479}{17824555783}a^{34}+\frac{156484915}{17824555783}a^{33}-\frac{215323807}{17824555783}a^{32}+\frac{133828080}{17824555783}a^{31}-\frac{41576864}{17824555783}a^{30}-\frac{57859059}{17824555783}a^{29}-\frac{99002794}{17824555783}a^{28}-\frac{199437683}{17824555783}a^{27}-\frac{18187798}{17824555783}a^{26}-\frac{112234610}{17824555783}a^{25}+\frac{1798800594}{17824555783}a^{24}-\frac{8179202140}{17824555783}a^{23}-\frac{570557111}{17824555783}a^{22}+\frac{3207405620}{17824555783}a^{21}-\frac{2942208648}{17824555783}a^{20}-\frac{2144551852}{17824555783}a^{19}-\frac{6701921604}{17824555783}a^{18}-\frac{2877292304}{17824555783}a^{17}+\frac{3389516452}{17824555783}a^{16}-\frac{1790881782}{17824555783}a^{15}-\frac{5204462427}{17824555783}a^{14}+\frac{4575109874}{17824555783}a^{13}+\frac{7536009740}{17824555783}a^{12}-\frac{323805813}{17824555783}a^{11}+\frac{8028821705}{17824555783}a^{10}+\frac{8791287601}{17824555783}a^{9}+\frac{4787688888}{17824555783}a^{8}-\frac{7551023635}{17824555783}a^{7}+\frac{595638208}{17824555783}a^{6}+\frac{8008195644}{17824555783}a^{5}+\frac{1313535806}{17824555783}a^{4}+\frac{3903536063}{17824555783}a^{3}+\frac{2428942905}{17824555783}a^{2}+\frac{173996133}{434745263}a-\frac{1507128}{10603543}$, $\frac{1}{20\!\cdots\!33}a^{40}-\frac{399683}{20\!\cdots\!33}a^{39}-\frac{826799221005}{20\!\cdots\!33}a^{38}-\frac{4336419085009}{20\!\cdots\!33}a^{37}+\frac{3691178725091}{20\!\cdots\!33}a^{36}+\frac{365444999491}{20\!\cdots\!33}a^{35}+\frac{107243113442230}{20\!\cdots\!33}a^{34}+\frac{171740365724889}{20\!\cdots\!33}a^{33}-\frac{238723489637266}{20\!\cdots\!33}a^{32}-\frac{174877874116048}{20\!\cdots\!33}a^{31}-\frac{210652815034695}{20\!\cdots\!33}a^{30}+\frac{222228038336903}{20\!\cdots\!33}a^{29}-\frac{118504331863883}{20\!\cdots\!33}a^{28}+\frac{14487213040783}{20\!\cdots\!33}a^{27}+\frac{33136697391676}{20\!\cdots\!33}a^{26}-\frac{84641449445192}{20\!\cdots\!33}a^{25}-\frac{76\!\cdots\!29}{20\!\cdots\!33}a^{24}+\frac{747487061362622}{20\!\cdots\!33}a^{23}+\frac{26967074923153}{20\!\cdots\!33}a^{22}-\frac{81\!\cdots\!31}{20\!\cdots\!33}a^{21}+\frac{905046453429709}{20\!\cdots\!33}a^{20}+\frac{84\!\cdots\!94}{20\!\cdots\!33}a^{19}+\frac{39\!\cdots\!57}{20\!\cdots\!33}a^{18}+\frac{234506402979862}{20\!\cdots\!33}a^{17}-\frac{84\!\cdots\!32}{20\!\cdots\!33}a^{16}+\frac{52\!\cdots\!12}{20\!\cdots\!33}a^{15}+\frac{30\!\cdots\!18}{20\!\cdots\!33}a^{14}-\frac{30\!\cdots\!12}{20\!\cdots\!33}a^{13}-\frac{12\!\cdots\!45}{20\!\cdots\!33}a^{12}-\frac{71\!\cdots\!96}{20\!\cdots\!33}a^{11}+\frac{47\!\cdots\!27}{20\!\cdots\!33}a^{10}-\frac{39\!\cdots\!82}{20\!\cdots\!33}a^{9}+\frac{79\!\cdots\!77}{20\!\cdots\!33}a^{8}-\frac{27\!\cdots\!86}{20\!\cdots\!33}a^{7}-\frac{62\!\cdots\!71}{20\!\cdots\!33}a^{6}+\frac{76\!\cdots\!32}{20\!\cdots\!33}a^{5}+\frac{56\!\cdots\!43}{20\!\cdots\!33}a^{4}+\frac{92\!\cdots\!97}{20\!\cdots\!33}a^{3}+\frac{60\!\cdots\!44}{20\!\cdots\!33}a^{2}+\frac{74926238270885}{498588039616813}a-\frac{2640854253287}{12160683893093}$, $\frac{1}{37\!\cdots\!41}a^{41}-\frac{49\!\cdots\!69}{37\!\cdots\!41}a^{40}-\frac{70\!\cdots\!83}{37\!\cdots\!41}a^{39}-\frac{80\!\cdots\!95}{37\!\cdots\!41}a^{38}-\frac{25\!\cdots\!38}{37\!\cdots\!41}a^{37}+\frac{33\!\cdots\!26}{37\!\cdots\!41}a^{36}+\frac{33\!\cdots\!97}{37\!\cdots\!41}a^{35}+\frac{45\!\cdots\!26}{37\!\cdots\!41}a^{34}-\frac{73\!\cdots\!81}{37\!\cdots\!41}a^{33}+\frac{13\!\cdots\!37}{37\!\cdots\!41}a^{32}+\frac{44\!\cdots\!08}{37\!\cdots\!41}a^{31}+\frac{14\!\cdots\!16}{37\!\cdots\!41}a^{30}-\frac{17\!\cdots\!82}{92\!\cdots\!01}a^{29}-\frac{24\!\cdots\!64}{37\!\cdots\!41}a^{28}-\frac{44\!\cdots\!87}{37\!\cdots\!41}a^{27}+\frac{16\!\cdots\!32}{37\!\cdots\!41}a^{26}+\frac{36\!\cdots\!00}{37\!\cdots\!41}a^{25}-\frac{14\!\cdots\!58}{37\!\cdots\!41}a^{24}-\frac{97\!\cdots\!20}{37\!\cdots\!41}a^{23}-\frac{86\!\cdots\!76}{37\!\cdots\!41}a^{22}+\frac{55\!\cdots\!72}{37\!\cdots\!41}a^{21}+\frac{14\!\cdots\!38}{37\!\cdots\!41}a^{20}+\frac{13\!\cdots\!48}{37\!\cdots\!41}a^{19}-\frac{30\!\cdots\!86}{37\!\cdots\!41}a^{18}+\frac{64\!\cdots\!31}{37\!\cdots\!41}a^{17}-\frac{10\!\cdots\!17}{37\!\cdots\!41}a^{16}-\frac{39\!\cdots\!88}{37\!\cdots\!41}a^{15}-\frac{14\!\cdots\!48}{37\!\cdots\!41}a^{14}-\frac{94\!\cdots\!65}{37\!\cdots\!41}a^{13}+\frac{86\!\cdots\!71}{37\!\cdots\!41}a^{12}+\frac{16\!\cdots\!94}{37\!\cdots\!41}a^{11}+\frac{33\!\cdots\!22}{37\!\cdots\!41}a^{10}+\frac{18\!\cdots\!78}{37\!\cdots\!41}a^{9}-\frac{19\!\cdots\!34}{92\!\cdots\!01}a^{8}-\frac{14\!\cdots\!17}{37\!\cdots\!41}a^{7}+\frac{18\!\cdots\!62}{37\!\cdots\!41}a^{6}+\frac{16\!\cdots\!84}{37\!\cdots\!41}a^{5}-\frac{12\!\cdots\!22}{37\!\cdots\!41}a^{4}-\frac{22\!\cdots\!59}{37\!\cdots\!41}a^{3}+\frac{20\!\cdots\!86}{37\!\cdots\!41}a^{2}+\frac{37\!\cdots\!73}{92\!\cdots\!01}a-\frac{71\!\cdots\!98}{73\!\cdots\!23}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $41$

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:  $20$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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^42 - x^41 + 20*x^40 - 25*x^39 - 8*x^38 - 138*x^37 - 2077*x^36 + 839*x^35 - 7305*x^34 + 8805*x^33 + 67596*x^32 - 12107*x^31 + 444232*x^30 - 141646*x^29 + 238560*x^28 + 2150502*x^27 - 3285732*x^26 + 14361861*x^25 - 708679*x^24 + 4004385*x^23 + 63920534*x^22 - 73366557*x^21 + 232544983*x^20 + 114300646*x^19 + 134380032*x^18 + 644451241*x^17 - 110366019*x^16 + 50739485*x^15 + 3000292231*x^14 - 3874326960*x^13 + 7531027885*x^12 - 11845394651*x^11 + 8115584321*x^10 - 10398559445*x^9 + 11442787794*x^8 - 3901741530*x^7 + 3575301015*x^6 - 3614499453*x^5 - 715331685*x^4 + 331663193*x^3 + 2507361126*x^2 - 1452650295*x + 649728353)
 
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^42 - x^41 + 20*x^40 - 25*x^39 - 8*x^38 - 138*x^37 - 2077*x^36 + 839*x^35 - 7305*x^34 + 8805*x^33 + 67596*x^32 - 12107*x^31 + 444232*x^30 - 141646*x^29 + 238560*x^28 + 2150502*x^27 - 3285732*x^26 + 14361861*x^25 - 708679*x^24 + 4004385*x^23 + 63920534*x^22 - 73366557*x^21 + 232544983*x^20 + 114300646*x^19 + 134380032*x^18 + 644451241*x^17 - 110366019*x^16 + 50739485*x^15 + 3000292231*x^14 - 3874326960*x^13 + 7531027885*x^12 - 11845394651*x^11 + 8115584321*x^10 - 10398559445*x^9 + 11442787794*x^8 - 3901741530*x^7 + 3575301015*x^6 - 3614499453*x^5 - 715331685*x^4 + 331663193*x^3 + 2507361126*x^2 - 1452650295*x + 649728353, 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^42 - x^41 + 20*x^40 - 25*x^39 - 8*x^38 - 138*x^37 - 2077*x^36 + 839*x^35 - 7305*x^34 + 8805*x^33 + 67596*x^32 - 12107*x^31 + 444232*x^30 - 141646*x^29 + 238560*x^28 + 2150502*x^27 - 3285732*x^26 + 14361861*x^25 - 708679*x^24 + 4004385*x^23 + 63920534*x^22 - 73366557*x^21 + 232544983*x^20 + 114300646*x^19 + 134380032*x^18 + 644451241*x^17 - 110366019*x^16 + 50739485*x^15 + 3000292231*x^14 - 3874326960*x^13 + 7531027885*x^12 - 11845394651*x^11 + 8115584321*x^10 - 10398559445*x^9 + 11442787794*x^8 - 3901741530*x^7 + 3575301015*x^6 - 3614499453*x^5 - 715331685*x^4 + 331663193*x^3 + 2507361126*x^2 - 1452650295*x + 649728353);
 
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^42 - x^41 + 20*x^40 - 25*x^39 - 8*x^38 - 138*x^37 - 2077*x^36 + 839*x^35 - 7305*x^34 + 8805*x^33 + 67596*x^32 - 12107*x^31 + 444232*x^30 - 141646*x^29 + 238560*x^28 + 2150502*x^27 - 3285732*x^26 + 14361861*x^25 - 708679*x^24 + 4004385*x^23 + 63920534*x^22 - 73366557*x^21 + 232544983*x^20 + 114300646*x^19 + 134380032*x^18 + 644451241*x^17 - 110366019*x^16 + 50739485*x^15 + 3000292231*x^14 - 3874326960*x^13 + 7531027885*x^12 - 11845394651*x^11 + 8115584321*x^10 - 10398559445*x^9 + 11442787794*x^8 - 3901741530*x^7 + 3575301015*x^6 - 3614499453*x^5 - 715331685*x^4 + 331663193*x^3 + 2507361126*x^2 - 1452650295*x + 649728353);
 
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

$C_{42}$ (as 42T1):

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)
 
A cyclic group of order 42
The 42 conjugacy class representatives for $C_{42}$
Character table for $C_{42}$ is not computed

Intermediate fields

\(\Q(\sqrt{-203}) \), \(\Q(\zeta_{7})^+\), 6.0.409905923.1, 7.7.594823321.1, 14.0.8450068952156066122535627.1, 21.21.142736986105602839685204351151303673689.1

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 $42$ $21^{2}$ $42$ R $42$ ${\href{/padicField/13.14.0.1}{14} }^{3}$ ${\href{/padicField/17.3.0.1}{3} }^{14}$ $21^{2}$ $21^{2}$ R $21^{2}$ $42$ ${\href{/padicField/41.1.0.1}{1} }^{42}$ ${\href{/padicField/43.14.0.1}{14} }^{3}$ $21^{2}$ $21^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{7}$

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
\(7\) Copy content Toggle raw display Deg $42$$6$$7$$35$
\(29\) Copy content Toggle raw display 29.14.13.1$x^{14} + 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.1$x^{14} + 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.1$x^{14} + 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$