Properties

Label 44.0.500...601.1
Degree $44$
Signature $[0, 22]$
Discriminant $5.004\times 10^{81}$
Root discriminant $71.91$
Ramified primes $13, 23$
Class number not computed
Class group not computed
Galois group $C_2\times C_{22}$ (as 44T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 + 4*x^42 - 7*x^41 + 19*x^40 - 40*x^39 + 97*x^38 - 217*x^37 + 508*x^36 - 1159*x^35 + 2683*x^34 - 6160*x^33 + 14209*x^32 - 32689*x^31 + 75316*x^30 - 173383*x^29 + 399331*x^28 - 919480*x^27 + 2117473*x^26 - 4875913*x^25 + 11228332*x^24 - 25856071*x^23 + 59541067*x^22 + 77568213*x^21 + 101054988*x^20 + 131649651*x^19 + 171515313*x^18 + 223433640*x^17 + 291112299*x^16 + 379188621*x^15 + 494148276*x^14 + 643417587*x^13 + 839027241*x^12 + 1091225520*x^11 + 1425856203*x^10 + 1847820357*x^9 + 2429748252*x^8 + 3113712819*x^7 + 4175531937*x^6 + 5165606520*x^5 + 7360989291*x^4 + 8135830269*x^3 + 13947137604*x^2 + 10460353203*x + 31381059609)
 
gp: K = bnfinit(x^44 - x^43 + 4*x^42 - 7*x^41 + 19*x^40 - 40*x^39 + 97*x^38 - 217*x^37 + 508*x^36 - 1159*x^35 + 2683*x^34 - 6160*x^33 + 14209*x^32 - 32689*x^31 + 75316*x^30 - 173383*x^29 + 399331*x^28 - 919480*x^27 + 2117473*x^26 - 4875913*x^25 + 11228332*x^24 - 25856071*x^23 + 59541067*x^22 + 77568213*x^21 + 101054988*x^20 + 131649651*x^19 + 171515313*x^18 + 223433640*x^17 + 291112299*x^16 + 379188621*x^15 + 494148276*x^14 + 643417587*x^13 + 839027241*x^12 + 1091225520*x^11 + 1425856203*x^10 + 1847820357*x^9 + 2429748252*x^8 + 3113712819*x^7 + 4175531937*x^6 + 5165606520*x^5 + 7360989291*x^4 + 8135830269*x^3 + 13947137604*x^2 + 10460353203*x + 31381059609, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31381059609, 10460353203, 13947137604, 8135830269, 7360989291, 5165606520, 4175531937, 3113712819, 2429748252, 1847820357, 1425856203, 1091225520, 839027241, 643417587, 494148276, 379188621, 291112299, 223433640, 171515313, 131649651, 101054988, 77568213, 59541067, -25856071, 11228332, -4875913, 2117473, -919480, 399331, -173383, 75316, -32689, 14209, -6160, 2683, -1159, 508, -217, 97, -40, 19, -7, 4, -1, 1]);
 

\(x^{44} - x^{43} + 4 x^{42} - 7 x^{41} + 19 x^{40} - 40 x^{39} + 97 x^{38} - 217 x^{37} + 508 x^{36} - 1159 x^{35} + 2683 x^{34} - 6160 x^{33} + 14209 x^{32} - 32689 x^{31} + 75316 x^{30} - 173383 x^{29} + 399331 x^{28} - 919480 x^{27} + 2117473 x^{26} - 4875913 x^{25} + 11228332 x^{24} - 25856071 x^{23} + 59541067 x^{22} + 77568213 x^{21} + 101054988 x^{20} + 131649651 x^{19} + 171515313 x^{18} + 223433640 x^{17} + 291112299 x^{16} + 379188621 x^{15} + 494148276 x^{14} + 643417587 x^{13} + 839027241 x^{12} + 1091225520 x^{11} + 1425856203 x^{10} + 1847820357 x^{9} + 2429748252 x^{8} + 3113712819 x^{7} + 4175531937 x^{6} + 5165606520 x^{5} + 7360989291 x^{4} + 8135830269 x^{3} + 13947137604 x^{2} + 10460353203 x + 31381059609\)  Toggle raw display

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

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 22]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(500\!\cdots\!601\)\(\medspace = 13^{22}\cdot 23^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $71.91$
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:  $13, 23$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $44$
This field is Galois and abelian over $\Q$.
Conductor:  \(299=13\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{299}(1,·)$, $\chi_{299}(131,·)$, $\chi_{299}(261,·)$, $\chi_{299}(129,·)$, $\chi_{299}(12,·)$, $\chi_{299}(66,·)$, $\chi_{299}(142,·)$, $\chi_{299}(144,·)$, $\chi_{299}(274,·)$, $\chi_{299}(259,·)$, $\chi_{299}(25,·)$, $\chi_{299}(155,·)$, $\chi_{299}(285,·)$, $\chi_{299}(287,·)$, $\chi_{299}(27,·)$, $\chi_{299}(38,·)$, $\chi_{299}(40,·)$, $\chi_{299}(170,·)$, $\chi_{299}(157,·)$, $\chi_{299}(51,·)$, $\chi_{299}(53,·)$, $\chi_{299}(183,·)$, $\chi_{299}(181,·)$, $\chi_{299}(64,·)$, $\chi_{299}(194,·)$, $\chi_{299}(196,·)$, $\chi_{299}(118,·)$, $\chi_{299}(77,·)$, $\chi_{299}(79,·)$, $\chi_{299}(209,·)$, $\chi_{299}(14,·)$, $\chi_{299}(90,·)$, $\chi_{299}(220,·)$, $\chi_{299}(222,·)$, $\chi_{299}(272,·)$, $\chi_{299}(103,·)$, $\chi_{299}(105,·)$, $\chi_{299}(235,·)$, $\chi_{299}(168,·)$, $\chi_{299}(116,·)$, $\chi_{299}(246,·)$, $\chi_{299}(233,·)$, $\chi_{299}(248,·)$, $\chi_{299}(298,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{178623201} a^{23} - \frac{1}{3} a^{22} + \frac{1}{3} a^{21} - \frac{1}{3} a^{20} + \frac{1}{3} a^{19} - \frac{1}{3} a^{18} + \frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{25856071}{59541067}$, $\frac{1}{535869603} a^{24} - \frac{1}{535869603} a^{23} + \frac{4}{9} a^{22} + \frac{2}{9} a^{21} + \frac{1}{9} a^{20} - \frac{4}{9} a^{19} - \frac{2}{9} a^{18} - \frac{1}{9} a^{17} + \frac{4}{9} a^{16} + \frac{2}{9} a^{15} + \frac{1}{9} a^{14} - \frac{4}{9} a^{13} - \frac{2}{9} a^{12} - \frac{1}{9} a^{11} + \frac{4}{9} a^{10} + \frac{2}{9} a^{9} + \frac{1}{9} a^{8} - \frac{4}{9} a^{7} - \frac{2}{9} a^{6} - \frac{1}{9} a^{5} + \frac{4}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2} + \frac{25856071}{178623201} a + \frac{11228332}{59541067}$, $\frac{1}{1607608809} a^{25} - \frac{1}{1607608809} a^{24} + \frac{4}{1607608809} a^{23} + \frac{11}{27} a^{22} + \frac{1}{27} a^{21} + \frac{5}{27} a^{20} - \frac{2}{27} a^{19} - \frac{10}{27} a^{18} + \frac{4}{27} a^{17} - \frac{7}{27} a^{16} - \frac{8}{27} a^{15} - \frac{13}{27} a^{14} - \frac{11}{27} a^{13} - \frac{1}{27} a^{12} - \frac{5}{27} a^{11} + \frac{2}{27} a^{10} + \frac{10}{27} a^{9} - \frac{4}{27} a^{8} + \frac{7}{27} a^{7} + \frac{8}{27} a^{6} + \frac{13}{27} a^{5} + \frac{11}{27} a^{4} + \frac{1}{27} a^{3} + \frac{25856071}{535869603} a^{2} + \frac{11228332}{178623201} a + \frac{4875913}{59541067}$, $\frac{1}{4822826427} a^{26} - \frac{1}{4822826427} a^{25} + \frac{4}{4822826427} a^{24} - \frac{7}{4822826427} a^{23} + \frac{28}{81} a^{22} + \frac{5}{81} a^{21} - \frac{2}{81} a^{20} + \frac{17}{81} a^{19} - \frac{23}{81} a^{18} - \frac{7}{81} a^{17} + \frac{19}{81} a^{16} - \frac{40}{81} a^{15} + \frac{16}{81} a^{14} + \frac{26}{81} a^{13} + \frac{22}{81} a^{12} - \frac{25}{81} a^{11} + \frac{10}{81} a^{10} - \frac{4}{81} a^{9} + \frac{34}{81} a^{8} + \frac{35}{81} a^{7} - \frac{14}{81} a^{6} + \frac{38}{81} a^{5} + \frac{1}{81} a^{4} + \frac{25856071}{1607608809} a^{3} + \frac{11228332}{535869603} a^{2} + \frac{4875913}{178623201} a + \frac{2117473}{59541067}$, $\frac{1}{14468479281} a^{27} - \frac{1}{14468479281} a^{26} + \frac{4}{14468479281} a^{25} - \frac{7}{14468479281} a^{24} + \frac{19}{14468479281} a^{23} + \frac{5}{243} a^{22} + \frac{79}{243} a^{21} - \frac{64}{243} a^{20} + \frac{58}{243} a^{19} - \frac{7}{243} a^{18} - \frac{62}{243} a^{17} + \frac{41}{243} a^{16} + \frac{16}{243} a^{15} + \frac{107}{243} a^{14} - \frac{59}{243} a^{13} - \frac{106}{243} a^{12} - \frac{71}{243} a^{11} - \frac{4}{243} a^{10} + \frac{34}{243} a^{9} - \frac{46}{243} a^{8} - \frac{95}{243} a^{7} - \frac{43}{243} a^{6} + \frac{1}{243} a^{5} + \frac{25856071}{4822826427} a^{4} + \frac{11228332}{1607608809} a^{3} + \frac{4875913}{535869603} a^{2} + \frac{2117473}{178623201} a + \frac{919480}{59541067}$, $\frac{1}{43405437843} a^{28} - \frac{1}{43405437843} a^{27} + \frac{4}{43405437843} a^{26} - \frac{7}{43405437843} a^{25} + \frac{19}{43405437843} a^{24} - \frac{40}{43405437843} a^{23} + \frac{79}{729} a^{22} - \frac{64}{729} a^{21} + \frac{301}{729} a^{20} + \frac{236}{729} a^{19} - \frac{62}{729} a^{18} + \frac{41}{729} a^{17} - \frac{227}{729} a^{16} + \frac{350}{729} a^{15} - \frac{302}{729} a^{14} - \frac{106}{729} a^{13} - \frac{71}{729} a^{12} - \frac{247}{729} a^{11} + \frac{34}{729} a^{10} - \frac{46}{729} a^{9} + \frac{148}{729} a^{8} - \frac{286}{729} a^{7} + \frac{1}{729} a^{6} + \frac{25856071}{14468479281} a^{5} + \frac{11228332}{4822826427} a^{4} + \frac{4875913}{1607608809} a^{3} + \frac{2117473}{535869603} a^{2} + \frac{919480}{178623201} a + \frac{399331}{59541067}$, $\frac{1}{130216313529} a^{29} - \frac{1}{130216313529} a^{28} + \frac{4}{130216313529} a^{27} - \frac{7}{130216313529} a^{26} + \frac{19}{130216313529} a^{25} - \frac{40}{130216313529} a^{24} + \frac{97}{130216313529} a^{23} - \frac{793}{2187} a^{22} + \frac{1030}{2187} a^{21} + \frac{965}{2187} a^{20} - \frac{62}{2187} a^{19} + \frac{770}{2187} a^{18} - \frac{956}{2187} a^{17} + \frac{1079}{2187} a^{16} + \frac{427}{2187} a^{15} + \frac{623}{2187} a^{14} + \frac{658}{2187} a^{13} - \frac{976}{2187} a^{12} + \frac{763}{2187} a^{11} + \frac{683}{2187} a^{10} - \frac{581}{2187} a^{9} + \frac{443}{2187} a^{8} + \frac{1}{2187} a^{7} + \frac{25856071}{43405437843} a^{6} + \frac{11228332}{14468479281} a^{5} + \frac{4875913}{4822826427} a^{4} + \frac{2117473}{1607608809} a^{3} + \frac{919480}{535869603} a^{2} + \frac{399331}{178623201} a + \frac{173383}{59541067}$, $\frac{1}{390648940587} a^{30} - \frac{1}{390648940587} a^{29} + \frac{4}{390648940587} a^{28} - \frac{7}{390648940587} a^{27} + \frac{19}{390648940587} a^{26} - \frac{40}{390648940587} a^{25} + \frac{97}{390648940587} a^{24} - \frac{217}{390648940587} a^{23} + \frac{3217}{6561} a^{22} + \frac{965}{6561} a^{21} + \frac{2125}{6561} a^{20} + \frac{770}{6561} a^{19} - \frac{956}{6561} a^{18} + \frac{3266}{6561} a^{17} + \frac{427}{6561} a^{16} + \frac{2810}{6561} a^{15} - \frac{1529}{6561} a^{14} - \frac{3163}{6561} a^{13} - \frac{1424}{6561} a^{12} - \frac{1504}{6561} a^{11} - \frac{2768}{6561} a^{10} - \frac{1744}{6561} a^{9} + \frac{1}{6561} a^{8} + \frac{25856071}{130216313529} a^{7} + \frac{11228332}{43405437843} a^{6} + \frac{4875913}{14468479281} a^{5} + \frac{2117473}{4822826427} a^{4} + \frac{919480}{1607608809} a^{3} + \frac{399331}{535869603} a^{2} + \frac{173383}{178623201} a + \frac{75316}{59541067}$, $\frac{1}{1171946821761} a^{31} - \frac{1}{1171946821761} a^{30} + \frac{4}{1171946821761} a^{29} - \frac{7}{1171946821761} a^{28} + \frac{19}{1171946821761} a^{27} - \frac{40}{1171946821761} a^{26} + \frac{97}{1171946821761} a^{25} - \frac{217}{1171946821761} a^{24} + \frac{508}{1171946821761} a^{23} + \frac{7526}{19683} a^{22} + \frac{2125}{19683} a^{21} + \frac{770}{19683} a^{20} + \frac{5605}{19683} a^{19} - \frac{3295}{19683} a^{18} + \frac{427}{19683} a^{17} + \frac{9371}{19683} a^{16} - \frac{8090}{19683} a^{15} - \frac{3163}{19683} a^{14} - \frac{1424}{19683} a^{13} - \frac{8065}{19683} a^{12} + \frac{3793}{19683} a^{11} - \frac{8305}{19683} a^{10} + \frac{1}{19683} a^{9} + \frac{25856071}{390648940587} a^{8} + \frac{11228332}{130216313529} a^{7} + \frac{4875913}{43405437843} a^{6} + \frac{2117473}{14468479281} a^{5} + \frac{919480}{4822826427} a^{4} + \frac{399331}{1607608809} a^{3} + \frac{173383}{535869603} a^{2} + \frac{75316}{178623201} a + \frac{32689}{59541067}$, $\frac{1}{3515840465283} a^{32} - \frac{1}{3515840465283} a^{31} + \frac{4}{3515840465283} a^{30} - \frac{7}{3515840465283} a^{29} + \frac{19}{3515840465283} a^{28} - \frac{40}{3515840465283} a^{27} + \frac{97}{3515840465283} a^{26} - \frac{217}{3515840465283} a^{25} + \frac{508}{3515840465283} a^{24} - \frac{1159}{3515840465283} a^{23} + \frac{2125}{59049} a^{22} + \frac{20453}{59049} a^{21} - \frac{14078}{59049} a^{20} + \frac{16388}{59049} a^{19} + \frac{427}{59049} a^{18} - \frac{10312}{59049} a^{17} + \frac{11593}{59049} a^{16} + \frac{16520}{59049} a^{15} + \frac{18259}{59049} a^{14} - \frac{27748}{59049} a^{13} + \frac{23476}{59049} a^{12} + \frac{11378}{59049} a^{11} + \frac{1}{59049} a^{10} + \frac{25856071}{1171946821761} a^{9} + \frac{11228332}{390648940587} a^{8} + \frac{4875913}{130216313529} a^{7} + \frac{2117473}{43405437843} a^{6} + \frac{919480}{14468479281} a^{5} + \frac{399331}{4822826427} a^{4} + \frac{173383}{1607608809} a^{3} + \frac{75316}{535869603} a^{2} + \frac{32689}{178623201} a + \frac{14209}{59541067}$, $\frac{1}{10547521395849} a^{33} - \frac{1}{10547521395849} a^{32} + \frac{4}{10547521395849} a^{31} - \frac{7}{10547521395849} a^{30} + \frac{19}{10547521395849} a^{29} - \frac{40}{10547521395849} a^{28} + \frac{97}{10547521395849} a^{27} - \frac{217}{10547521395849} a^{26} + \frac{508}{10547521395849} a^{25} - \frac{1159}{10547521395849} a^{24} + \frac{2683}{10547521395849} a^{23} + \frac{79502}{177147} a^{22} - \frac{73127}{177147} a^{21} - \frac{42661}{177147} a^{20} + \frac{427}{177147} a^{19} + \frac{48737}{177147} a^{18} - \frac{47456}{177147} a^{17} + \frac{16520}{177147} a^{16} + \frac{18259}{177147} a^{15} + \frac{31301}{177147} a^{14} + \frac{23476}{177147} a^{13} + \frac{70427}{177147} a^{12} + \frac{1}{177147} a^{11} + \frac{25856071}{3515840465283} a^{10} + \frac{11228332}{1171946821761} a^{9} + \frac{4875913}{390648940587} a^{8} + \frac{2117473}{130216313529} a^{7} + \frac{919480}{43405437843} a^{6} + \frac{399331}{14468479281} a^{5} + \frac{173383}{4822826427} a^{4} + \frac{75316}{1607608809} a^{3} + \frac{32689}{535869603} a^{2} + \frac{14209}{178623201} a + \frac{6160}{59541067}$, $\frac{1}{31642564187547} a^{34} - \frac{1}{31642564187547} a^{33} + \frac{4}{31642564187547} a^{32} - \frac{7}{31642564187547} a^{31} + \frac{19}{31642564187547} a^{30} - \frac{40}{31642564187547} a^{29} + \frac{97}{31642564187547} a^{28} - \frac{217}{31642564187547} a^{27} + \frac{508}{31642564187547} a^{26} - \frac{1159}{31642564187547} a^{25} + \frac{2683}{31642564187547} a^{24} - \frac{6160}{31642564187547} a^{23} + \frac{104020}{531441} a^{22} + \frac{134486}{531441} a^{21} + \frac{177574}{531441} a^{20} + \frac{225884}{531441} a^{19} - \frac{224603}{531441} a^{18} - \frac{160627}{531441} a^{17} + \frac{18259}{531441} a^{16} + \frac{31301}{531441} a^{15} + \frac{23476}{531441} a^{14} + \frac{70427}{531441} a^{13} + \frac{1}{531441} a^{12} + \frac{25856071}{10547521395849} a^{11} + \frac{11228332}{3515840465283} a^{10} + \frac{4875913}{1171946821761} a^{9} + \frac{2117473}{390648940587} a^{8} + \frac{919480}{130216313529} a^{7} + \frac{399331}{43405437843} a^{6} + \frac{173383}{14468479281} a^{5} + \frac{75316}{4822826427} a^{4} + \frac{32689}{1607608809} a^{3} + \frac{14209}{535869603} a^{2} + \frac{6160}{178623201} a + \frac{2683}{59541067}$, $\frac{1}{94927692562641} a^{35} - \frac{1}{94927692562641} a^{34} + \frac{4}{94927692562641} a^{33} - \frac{7}{94927692562641} a^{32} + \frac{19}{94927692562641} a^{31} - \frac{40}{94927692562641} a^{30} + \frac{97}{94927692562641} a^{29} - \frac{217}{94927692562641} a^{28} + \frac{508}{94927692562641} a^{27} - \frac{1159}{94927692562641} a^{26} + \frac{2683}{94927692562641} a^{25} - \frac{6160}{94927692562641} a^{24} + \frac{14209}{94927692562641} a^{23} - \frac{396955}{1594323} a^{22} + \frac{709015}{1594323} a^{21} - \frac{305557}{1594323} a^{20} - \frac{756044}{1594323} a^{19} - \frac{160627}{1594323} a^{18} - \frac{513182}{1594323} a^{17} + \frac{31301}{1594323} a^{16} + \frac{23476}{1594323} a^{15} + \frac{70427}{1594323} a^{14} + \frac{1}{1594323} a^{13} + \frac{25856071}{31642564187547} a^{12} + \frac{11228332}{10547521395849} a^{11} + \frac{4875913}{3515840465283} a^{10} + \frac{2117473}{1171946821761} a^{9} + \frac{919480}{390648940587} a^{8} + \frac{399331}{130216313529} a^{7} + \frac{173383}{43405437843} a^{6} + \frac{75316}{14468479281} a^{5} + \frac{32689}{4822826427} a^{4} + \frac{14209}{1607608809} a^{3} + \frac{6160}{535869603} a^{2} + \frac{2683}{178623201} a + \frac{1159}{59541067}$, $\frac{1}{284783077687923} a^{36} - \frac{1}{284783077687923} a^{35} + \frac{4}{284783077687923} a^{34} - \frac{7}{284783077687923} a^{33} + \frac{19}{284783077687923} a^{32} - \frac{40}{284783077687923} a^{31} + \frac{97}{284783077687923} a^{30} - \frac{217}{284783077687923} a^{29} + \frac{508}{284783077687923} a^{28} - \frac{1159}{284783077687923} a^{27} + \frac{2683}{284783077687923} a^{26} - \frac{6160}{284783077687923} a^{25} + \frac{14209}{284783077687923} a^{24} - \frac{32689}{284783077687923} a^{23} - \frac{885308}{4782969} a^{22} - \frac{305557}{4782969} a^{21} - \frac{2350367}{4782969} a^{20} + \frac{1433696}{4782969} a^{19} + \frac{1081141}{4782969} a^{18} - \frac{1563022}{4782969} a^{17} + \frac{23476}{4782969} a^{16} + \frac{70427}{4782969} a^{15} + \frac{1}{4782969} a^{14} + \frac{25856071}{94927692562641} a^{13} + \frac{11228332}{31642564187547} a^{12} + \frac{4875913}{10547521395849} a^{11} + \frac{2117473}{3515840465283} a^{10} + \frac{919480}{1171946821761} a^{9} + \frac{399331}{390648940587} a^{8} + \frac{173383}{130216313529} a^{7} + \frac{75316}{43405437843} a^{6} + \frac{32689}{14468479281} a^{5} + \frac{14209}{4822826427} a^{4} + \frac{6160}{1607608809} a^{3} + \frac{2683}{535869603} a^{2} + \frac{1159}{178623201} a + \frac{508}{59541067}$, $\frac{1}{854349233063769} a^{37} - \frac{1}{854349233063769} a^{36} + \frac{4}{854349233063769} a^{35} - \frac{7}{854349233063769} a^{34} + \frac{19}{854349233063769} a^{33} - \frac{40}{854349233063769} a^{32} + \frac{97}{854349233063769} a^{31} - \frac{217}{854349233063769} a^{30} + \frac{508}{854349233063769} a^{29} - \frac{1159}{854349233063769} a^{28} + \frac{2683}{854349233063769} a^{27} - \frac{6160}{854349233063769} a^{26} + \frac{14209}{854349233063769} a^{25} - \frac{32689}{854349233063769} a^{24} + \frac{75316}{854349233063769} a^{23} - \frac{5088526}{14348907} a^{22} + \frac{2432602}{14348907} a^{21} - \frac{3349273}{14348907} a^{20} - \frac{3701828}{14348907} a^{19} - \frac{6345991}{14348907} a^{18} - \frac{4759493}{14348907} a^{17} + \frac{70427}{14348907} a^{16} + \frac{1}{14348907} a^{15} + \frac{25856071}{284783077687923} a^{14} + \frac{11228332}{94927692562641} a^{13} + \frac{4875913}{31642564187547} a^{12} + \frac{2117473}{10547521395849} a^{11} + \frac{919480}{3515840465283} a^{10} + \frac{399331}{1171946821761} a^{9} + \frac{173383}{390648940587} a^{8} + \frac{75316}{130216313529} a^{7} + \frac{32689}{43405437843} a^{6} + \frac{14209}{14468479281} a^{5} + \frac{6160}{4822826427} a^{4} + \frac{2683}{1607608809} a^{3} + \frac{1159}{535869603} a^{2} + \frac{508}{178623201} a + \frac{217}{59541067}$, $\frac{1}{2563047699191307} a^{38} - \frac{1}{2563047699191307} a^{37} + \frac{4}{2563047699191307} a^{36} - \frac{7}{2563047699191307} a^{35} + \frac{19}{2563047699191307} a^{34} - \frac{40}{2563047699191307} a^{33} + \frac{97}{2563047699191307} a^{32} - \frac{217}{2563047699191307} a^{31} + \frac{508}{2563047699191307} a^{30} - \frac{1159}{2563047699191307} a^{29} + \frac{2683}{2563047699191307} a^{28} - \frac{6160}{2563047699191307} a^{27} + \frac{14209}{2563047699191307} a^{26} - \frac{32689}{2563047699191307} a^{25} + \frac{75316}{2563047699191307} a^{24} - \frac{173383}{2563047699191307} a^{23} + \frac{16781509}{43046721} a^{22} + \frac{10999634}{43046721} a^{21} - \frac{3701828}{43046721} a^{20} - \frac{6345991}{43046721} a^{19} - \frac{4759493}{43046721} a^{18} - \frac{14278480}{43046721} a^{17} + \frac{1}{43046721} a^{16} + \frac{25856071}{854349233063769} a^{15} + \frac{11228332}{284783077687923} a^{14} + \frac{4875913}{94927692562641} a^{13} + \frac{2117473}{31642564187547} a^{12} + \frac{919480}{10547521395849} a^{11} + \frac{399331}{3515840465283} a^{10} + \frac{173383}{1171946821761} a^{9} + \frac{75316}{390648940587} a^{8} + \frac{32689}{130216313529} a^{7} + \frac{14209}{43405437843} a^{6} + \frac{6160}{14468479281} a^{5} + \frac{2683}{4822826427} a^{4} + \frac{1159}{1607608809} a^{3} + \frac{508}{535869603} a^{2} + \frac{217}{178623201} a + \frac{97}{59541067}$, $\frac{1}{7689143097573921} a^{39} - \frac{1}{7689143097573921} a^{38} + \frac{4}{7689143097573921} a^{37} - \frac{7}{7689143097573921} a^{36} + \frac{19}{7689143097573921} a^{35} - \frac{40}{7689143097573921} a^{34} + \frac{97}{7689143097573921} a^{33} - \frac{217}{7689143097573921} a^{32} + \frac{508}{7689143097573921} a^{31} - \frac{1159}{7689143097573921} a^{30} + \frac{2683}{7689143097573921} a^{29} - \frac{6160}{7689143097573921} a^{28} + \frac{14209}{7689143097573921} a^{27} - \frac{32689}{7689143097573921} a^{26} + \frac{75316}{7689143097573921} a^{25} - \frac{173383}{7689143097573921} a^{24} + \frac{399331}{7689143097573921} a^{23} + \frac{10999634}{129140163} a^{22} + \frac{39344893}{129140163} a^{21} - \frac{6345991}{129140163} a^{20} - \frac{4759493}{129140163} a^{19} - \frac{14278480}{129140163} a^{18} + \frac{1}{129140163} a^{17} + \frac{25856071}{2563047699191307} a^{16} + \frac{11228332}{854349233063769} a^{15} + \frac{4875913}{284783077687923} a^{14} + \frac{2117473}{94927692562641} a^{13} + \frac{919480}{31642564187547} a^{12} + \frac{399331}{10547521395849} a^{11} + \frac{173383}{3515840465283} a^{10} + \frac{75316}{1171946821761} a^{9} + \frac{32689}{390648940587} a^{8} + \frac{14209}{130216313529} a^{7} + \frac{6160}{43405437843} a^{6} + \frac{2683}{14468479281} a^{5} + \frac{1159}{4822826427} a^{4} + \frac{508}{1607608809} a^{3} + \frac{217}{535869603} a^{2} + \frac{97}{178623201} a + \frac{40}{59541067}$, $\frac{1}{23067429292721763} a^{40} - \frac{1}{23067429292721763} a^{39} + \frac{4}{23067429292721763} a^{38} - \frac{7}{23067429292721763} a^{37} + \frac{19}{23067429292721763} a^{36} - \frac{40}{23067429292721763} a^{35} + \frac{97}{23067429292721763} a^{34} - \frac{217}{23067429292721763} a^{33} + \frac{508}{23067429292721763} a^{32} - \frac{1159}{23067429292721763} a^{31} + \frac{2683}{23067429292721763} a^{30} - \frac{6160}{23067429292721763} a^{29} + \frac{14209}{23067429292721763} a^{28} - \frac{32689}{23067429292721763} a^{27} + \frac{75316}{23067429292721763} a^{26} - \frac{173383}{23067429292721763} a^{25} + \frac{399331}{23067429292721763} a^{24} - \frac{919480}{23067429292721763} a^{23} - \frac{89795270}{387420489} a^{22} + \frac{122794172}{387420489} a^{21} - \frac{4759493}{387420489} a^{20} - \frac{14278480}{387420489} a^{19} + \frac{1}{387420489} a^{18} + \frac{25856071}{7689143097573921} a^{17} + \frac{11228332}{2563047699191307} a^{16} + \frac{4875913}{854349233063769} a^{15} + \frac{2117473}{284783077687923} a^{14} + \frac{919480}{94927692562641} a^{13} + \frac{399331}{31642564187547} a^{12} + \frac{173383}{10547521395849} a^{11} + \frac{75316}{3515840465283} a^{10} + \frac{32689}{1171946821761} a^{9} + \frac{14209}{390648940587} a^{8} + \frac{6160}{130216313529} a^{7} + \frac{2683}{43405437843} a^{6} + \frac{1159}{14468479281} a^{5} + \frac{508}{4822826427} a^{4} + \frac{217}{1607608809} a^{3} + \frac{97}{535869603} a^{2} + \frac{40}{178623201} a + \frac{19}{59541067}$, $\frac{1}{69202287878165289} a^{41} - \frac{1}{69202287878165289} a^{40} + \frac{4}{69202287878165289} a^{39} - \frac{7}{69202287878165289} a^{38} + \frac{19}{69202287878165289} a^{37} - \frac{40}{69202287878165289} a^{36} + \frac{97}{69202287878165289} a^{35} - \frac{217}{69202287878165289} a^{34} + \frac{508}{69202287878165289} a^{33} - \frac{1159}{69202287878165289} a^{32} + \frac{2683}{69202287878165289} a^{31} - \frac{6160}{69202287878165289} a^{30} + \frac{14209}{69202287878165289} a^{29} - \frac{32689}{69202287878165289} a^{28} + \frac{75316}{69202287878165289} a^{27} - \frac{173383}{69202287878165289} a^{26} + \frac{399331}{69202287878165289} a^{25} - \frac{919480}{69202287878165289} a^{24} + \frac{2117473}{69202287878165289} a^{23} + \frac{510214661}{1162261467} a^{22} + \frac{382660996}{1162261467} a^{21} - \frac{14278480}{1162261467} a^{20} + \frac{1}{1162261467} a^{19} + \frac{25856071}{23067429292721763} a^{18} + \frac{11228332}{7689143097573921} a^{17} + \frac{4875913}{2563047699191307} a^{16} + \frac{2117473}{854349233063769} a^{15} + \frac{919480}{284783077687923} a^{14} + \frac{399331}{94927692562641} a^{13} + \frac{173383}{31642564187547} a^{12} + \frac{75316}{10547521395849} a^{11} + \frac{32689}{3515840465283} a^{10} + \frac{14209}{1171946821761} a^{9} + \frac{6160}{390648940587} a^{8} + \frac{2683}{130216313529} a^{7} + \frac{1159}{43405437843} a^{6} + \frac{508}{14468479281} a^{5} + \frac{217}{4822826427} a^{4} + \frac{97}{1607608809} a^{3} + \frac{40}{535869603} a^{2} + \frac{19}{178623201} a + \frac{7}{59541067}$, $\frac{1}{207606863634495867} a^{42} - \frac{1}{207606863634495867} a^{41} + \frac{4}{207606863634495867} a^{40} - \frac{7}{207606863634495867} a^{39} + \frac{19}{207606863634495867} a^{38} - \frac{40}{207606863634495867} a^{37} + \frac{97}{207606863634495867} a^{36} - \frac{217}{207606863634495867} a^{35} + \frac{508}{207606863634495867} a^{34} - \frac{1159}{207606863634495867} a^{33} + \frac{2683}{207606863634495867} a^{32} - \frac{6160}{207606863634495867} a^{31} + \frac{14209}{207606863634495867} a^{30} - \frac{32689}{207606863634495867} a^{29} + \frac{75316}{207606863634495867} a^{28} - \frac{173383}{207606863634495867} a^{27} + \frac{399331}{207606863634495867} a^{26} - \frac{919480}{207606863634495867} a^{25} + \frac{2117473}{207606863634495867} a^{24} - \frac{4875913}{207606863634495867} a^{23} + \frac{382660996}{3486784401} a^{22} + \frac{1147982987}{3486784401} a^{21} + \frac{1}{3486784401} a^{20} + \frac{25856071}{69202287878165289} a^{19} + \frac{11228332}{23067429292721763} a^{18} + \frac{4875913}{7689143097573921} a^{17} + \frac{2117473}{2563047699191307} a^{16} + \frac{919480}{854349233063769} a^{15} + \frac{399331}{284783077687923} a^{14} + \frac{173383}{94927692562641} a^{13} + \frac{75316}{31642564187547} a^{12} + \frac{32689}{10547521395849} a^{11} + \frac{14209}{3515840465283} a^{10} + \frac{6160}{1171946821761} a^{9} + \frac{2683}{390648940587} a^{8} + \frac{1159}{130216313529} a^{7} + \frac{508}{43405437843} a^{6} + \frac{217}{14468479281} a^{5} + \frac{97}{4822826427} a^{4} + \frac{40}{1607608809} a^{3} + \frac{19}{535869603} a^{2} + \frac{7}{178623201} a + \frac{4}{59541067}$, $\frac{1}{622820590903487601} a^{43} - \frac{1}{622820590903487601} a^{42} + \frac{4}{622820590903487601} a^{41} - \frac{7}{622820590903487601} a^{40} + \frac{19}{622820590903487601} a^{39} - \frac{40}{622820590903487601} a^{38} + \frac{97}{622820590903487601} a^{37} - \frac{217}{622820590903487601} a^{36} + \frac{508}{622820590903487601} a^{35} - \frac{1159}{622820590903487601} a^{34} + \frac{2683}{622820590903487601} a^{33} - \frac{6160}{622820590903487601} a^{32} + \frac{14209}{622820590903487601} a^{31} - \frac{32689}{622820590903487601} a^{30} + \frac{75316}{622820590903487601} a^{29} - \frac{173383}{622820590903487601} a^{28} + \frac{399331}{622820590903487601} a^{27} - \frac{919480}{622820590903487601} a^{26} + \frac{2117473}{622820590903487601} a^{25} - \frac{4875913}{622820590903487601} a^{24} + \frac{11228332}{622820590903487601} a^{23} + \frac{1147982987}{10460353203} a^{22} + \frac{1}{10460353203} a^{21} + \frac{25856071}{207606863634495867} a^{20} + \frac{11228332}{69202287878165289} a^{19} + \frac{4875913}{23067429292721763} a^{18} + \frac{2117473}{7689143097573921} a^{17} + \frac{919480}{2563047699191307} a^{16} + \frac{399331}{854349233063769} a^{15} + \frac{173383}{284783077687923} a^{14} + \frac{75316}{94927692562641} a^{13} + \frac{32689}{31642564187547} a^{12} + \frac{14209}{10547521395849} a^{11} + \frac{6160}{3515840465283} a^{10} + \frac{2683}{1171946821761} a^{9} + \frac{1159}{390648940587} a^{8} + \frac{508}{130216313529} a^{7} + \frac{217}{43405437843} a^{6} + \frac{97}{14468479281} a^{5} + \frac{40}{4822826427} a^{4} + \frac{19}{1607608809} a^{3} + \frac{7}{535869603} a^{2} + \frac{4}{178623201} a + \frac{1}{59541067}$  Toggle raw display

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:  $21$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{399331}{7689143097573921} a^{40} - \frac{85722212350663}{7689143097573921} a^{17} \) (order $46$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed  Toggle raw display
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

$C_2\times C_{22}$ (as 44T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
An abelian group of order 44
The 44 conjugacy class representatives for $C_2\times C_{22}$
Character table for $C_2\times C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-299}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{13}, \sqrt{-23})\), \(\Q(\zeta_{23})^+\), 22.0.70739409751010214154180397092922226278051.1, \(\Q(\zeta_{23})\), 22.22.3075626510913487571920886830127053316437.1

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 $22^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ $22^{2}$ R $22^{2}$ $22^{2}$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$ $22^{2}$ $22^{2}$

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
13Data not computed
23Data not computed