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)

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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(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)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![649728353, -1452650295, 2507361126, 331663193, -715331685, -3614499453, 3575301015, -3901741530, 11442787794, -10398559445, 8115584321, -11845394651, 7531027885, -3874326960, 3000292231, 50739485, -110366019, 644451241, 134380032, 114300646, 232544983, -73366557, 63920534, 4004385, -708679, 14361861, -3285732, 2150502, 238560, -141646, 444232, -12107, 67596, 8805, -7305, 839, -2077, -138, -8, -25, 20, -1, 1]);
 

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

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 21]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-40\!\cdots\!667\)\(\medspace = -\,7^{35}\cdot 29^{39}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $115.40$
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:  $7, 29$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $42$
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.

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}{20442109624289333} a^{40} - \frac{399683}{20442109624289333} a^{39} - \frac{826799221005}{20442109624289333} a^{38} - \frac{4336419085009}{20442109624289333} a^{37} + \frac{3691178725091}{20442109624289333} a^{36} + \frac{365444999491}{20442109624289333} a^{35} + \frac{107243113442230}{20442109624289333} a^{34} + \frac{171740365724889}{20442109624289333} a^{33} - \frac{238723489637266}{20442109624289333} a^{32} - \frac{174877874116048}{20442109624289333} a^{31} - \frac{210652815034695}{20442109624289333} a^{30} + \frac{222228038336903}{20442109624289333} a^{29} - \frac{118504331863883}{20442109624289333} a^{28} + \frac{14487213040783}{20442109624289333} a^{27} + \frac{33136697391676}{20442109624289333} a^{26} - \frac{84641449445192}{20442109624289333} a^{25} - \frac{7632170212775429}{20442109624289333} a^{24} + \frac{747487061362622}{20442109624289333} a^{23} + \frac{26967074923153}{20442109624289333} a^{22} - \frac{8119478362934431}{20442109624289333} a^{21} + \frac{905046453429709}{20442109624289333} a^{20} + \frac{8416150061818094}{20442109624289333} a^{19} + \frac{3913446569713257}{20442109624289333} a^{18} + \frac{234506402979862}{20442109624289333} a^{17} - \frac{8430762221285432}{20442109624289333} a^{16} + \frac{5284520979377012}{20442109624289333} a^{15} + \frac{3019088995231118}{20442109624289333} a^{14} - \frac{3082560041798512}{20442109624289333} a^{13} - \frac{1292487273984645}{20442109624289333} a^{12} - \frac{7148726296719696}{20442109624289333} a^{11} + \frac{4786954734796927}{20442109624289333} a^{10} - \frac{3962517133865982}{20442109624289333} a^{9} + \frac{7929166046788177}{20442109624289333} a^{8} - \frac{2784789713019986}{20442109624289333} a^{7} - \frac{6286671406686871}{20442109624289333} a^{6} + \frac{7626811809479932}{20442109624289333} a^{5} + \frac{5618835758289543}{20442109624289333} a^{4} + \frac{9258770096393497}{20442109624289333} a^{3} + \frac{6052812711917444}{20442109624289333} a^{2} + \frac{74926238270885}{498588039616813} a - \frac{2640854253287}{12160683893093}$, $\frac{1}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{41} - \frac{49023333554475500810452083501847798007236938881351970226182415135621262998499681926740786623004938087277485261702782850068041357767908984005105231844593894805414528282690108095569}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{40} - \frac{70817692542750886201405393654845732666161866815630305582083961529645598757171128893016929054149830734740281182975305232632210835201311812463996828510609753868490230654054798909534769583}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{39} - \frac{801305254302782260572265689821372802523852909751070283862310656979981772515688389462166720906099891661306155550136289319688139973681442376184256635205061881329366091727188950003988257195222195}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{38} - \frac{258200234594931059857561490957243501416132125545269994461664944617781918296404318149274227715104646943021979679262208344555388197289062140510908910070795179548925300138910729403181661623497638}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{37} + \frac{332277949479167649867517809802227061200522403066021718074754809793476747080436929708607732115964983674605383998417905711782016654037877013775641571968682532121162590356672962368181236916961826}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{36} + \frac{333838185420290853730981324614907299523066080541109135016247406245124799510853476545099785693530029104649339338100134575108667992990953786707118237160247270985909773840838059258034970732663197}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{35} + \frac{4520101272693253206490492338416301868189593798452171665685189263767398058052896719521908688438665553733106726746023147425695159766929015769128740263550417875956163685378075487695054575762315626}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{34} - \frac{7342617590986118510028309745632454359812966614985567911696877492174165103543478399080344612310039704358739278835417161094213974554443020779859466977269612080067340678880206243272533178742630681}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{33} + \frac{13490403052242382102389861137467752476370078129728760735430518280756639102204094144634579618652297646735028435330609019157871103622430553546645692284288002764340063170700702922706579856186152737}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{32} + \frac{44126397996690203579815226308089652179030642897653462576537160832013765342954827814346669682652861652508348410735103011907763651432174861680616918842075407305387660199954486112791383650245615308}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{31} + \frac{14332282459150558452564538864570999346704215797762432128244165330714772639058327485146734800284951434458984701952088210844549954743272508051013879557184201269979075044855783227482514950588860216}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{30} - 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\frac{305719926255895996078458736049391070870418017465614333248668757684342624664533556439385771312606126431753397677162037328386586198251109144100681919166894779821967077177006048007395026594342230886}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{18} + \frac{646878910105394053756380423215382278840910740151481894519658691029795913744182491583875331116988263385551254643425444678149106070371378855966432406524204564880984834950777807615667965913738652831}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{17} - \frac{1040717674757922148989682852315163141958643898505046578757234197228270671783323086671061396458382598897908590744828562995061360519565449531660798556659704350034125345044403071861981269441206090517}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{16} - \frac{398651973131902991127080169803393145144036953161408263873381920532048493662398347858127352294378306127582855524243731450863333370844544171142138505537253843467407405493700167018637474342952774388}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{15} - \frac{1424039359667549836915201901508649484147553252025389443960083221882147751665427674822846792008501070898851043594354305095978809921029570148137922504673476402622942063499863079773661501272622921148}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{14} - \frac{945624231883743971016336025150143264395318668987289601075337227881608941297147991880822564265087294243803687012922588037951049334827403779689149223158319358799004425792578223233544397442312625965}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{13} + \frac{86840986914315369934765463832993069614109895696358964386873950634314266288683569711239268210350298494321513295806592188033982025804649534273989082066951677739606776116917193843981115993953416071}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{12} + \frac{1637255377993400234766299611439330338983576321727632184219545688755691206195435445868011329209401133524018428835593998301722754763827988865959693829368563327741115269713180896558562935185503098994}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{11} + \frac{33207331177831369618377742754579854474878358525175233256382046420581221040099490287882438771049660667134232844374442153683260097126901068876769129604780792321266776897824327662651206743128818222}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{10} + \frac{1849482267700834305995647802494446413459622822594204519847421917571029423277873415538987153883249282220174674150379710840572824512903044798779146956878779963029759770897162477609202199636445140678}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{9} - \frac{19850976136878184812941165806405282023629624921894296264701502272832613234384888658166842653646911860176959278975239961765824473145441544286801075220851664251718847129400833274362287803064764034}{92075994718102950030459812779591425256569783844744340468483759836364390291952704712820227377260412593697569678028099795469749322292352007498978101551988303957289573840595960712087696221633776901} a^{8} - \frac{1498687268948728272279058689663533452847635932178077469088265091169213536717691256197467743615652423108401152026743180901556997362749898083375702316523320372515913682069614659335289264818982431817}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{7} + \frac{185861143816126306550489757242933391747003025768792807972431794777120203844012356140058666433684448831188684522180625332915773497366573071259169760445232994488902474581305487040444507403456340262}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{6} + \frac{167631156759851652369972869565595622030815201720235564313815801788605973369226766334668148405198885559493898315230848319434682152509700465379142557246125373743691975800838130152020578329046154584}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{5} - \frac{1209160058088204136130559784199022059800338893295325343463655791539267025315578747498994045295741361036097966584000645283229935416659261932364991785878992317993063516406575507418899906465554705222}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{4} - \frac{220582559362367442634074645693412211506401902843158733174500679527040187359035937894052392967580240489019389173100071851222692579783354947688335510719899475429057371849742117765052526248202831459}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{3} + \frac{20941877144814611230062244582681779840235800447839478550980487635011265679850251403913861295922203093268065241570794832927596043579095808904337713862651764053333831205419764738803450377209852686}{3775115783442220951248852323963248435519361137634517959207834153290940001970060893225629322467676916341600356799152091614259722213986432307458102163631520462248872527464434389195595545086984852941} a^{2} + \frac{37727084729035782237809245419093942433555477468208356146884981202466723539442274952960826514362786513972373582826792501036017555962171991978394169819772992654150906952389554887751338734557683973}{92075994718102950030459812779591425256569783844744340468483759836364390291952704712820227377260412593697569678028099795469749322292352007498978101551988303957289573840595960712087696221633776901} a - \frac{716070836773309587387213304958434286077074352599371558346173099701333304656970203156445750141138921156274801724699672561414785512586416633177535294305759479606000431964970542191527287873298}{7315166021935564473699834176498881803175481357332512947365040107759147556363923469676668576885708476499369959325343592235620030371999047231189171490584595531682654631015806841351211267310223}$

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

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $20$
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:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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.

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

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

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
7Data not computed
$29$29.14.13.1$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}$