Properties

Label 44.0.126...409.1
Degree $44$
Signature $[0, 22]$
Discriminant $1.268\times 10^{80}$
Root discriminant $66.15$
Ramified primes $11, 23$
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 - 2*x^42 + 5*x^41 + x^40 - 16*x^39 + 13*x^38 + 35*x^37 - 74*x^36 - 31*x^35 + 253*x^34 - 160*x^33 - 599*x^32 + 1079*x^31 + 718*x^30 - 3955*x^29 + 1801*x^28 + 10064*x^27 - 15467*x^26 - 14725*x^25 + 61126*x^24 - 16951*x^23 - 166427*x^22 - 50853*x^21 + 550134*x^20 - 397575*x^19 - 1252827*x^18 + 2445552*x^17 + 1312929*x^16 - 8649585*x^15 + 4710798*x^14 + 21237957*x^13 - 35370351*x^12 - 28343520*x^11 + 134454573*x^10 - 49424013*x^9 - 353939706*x^8 + 502211745*x^7 + 559607373*x^6 - 2066242608*x^5 + 387420489*x^4 + 5811307335*x^3 - 6973568802*x^2 - 10460353203*x + 31381059609)
 
gp: K = bnfinit(x^44 - x^43 - 2*x^42 + 5*x^41 + x^40 - 16*x^39 + 13*x^38 + 35*x^37 - 74*x^36 - 31*x^35 + 253*x^34 - 160*x^33 - 599*x^32 + 1079*x^31 + 718*x^30 - 3955*x^29 + 1801*x^28 + 10064*x^27 - 15467*x^26 - 14725*x^25 + 61126*x^24 - 16951*x^23 - 166427*x^22 - 50853*x^21 + 550134*x^20 - 397575*x^19 - 1252827*x^18 + 2445552*x^17 + 1312929*x^16 - 8649585*x^15 + 4710798*x^14 + 21237957*x^13 - 35370351*x^12 - 28343520*x^11 + 134454573*x^10 - 49424013*x^9 - 353939706*x^8 + 502211745*x^7 + 559607373*x^6 - 2066242608*x^5 + 387420489*x^4 + 5811307335*x^3 - 6973568802*x^2 - 10460353203*x + 31381059609, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31381059609, -10460353203, -6973568802, 5811307335, 387420489, -2066242608, 559607373, 502211745, -353939706, -49424013, 134454573, -28343520, -35370351, 21237957, 4710798, -8649585, 1312929, 2445552, -1252827, -397575, 550134, -50853, -166427, -16951, 61126, -14725, -15467, 10064, 1801, -3955, 718, 1079, -599, -160, 253, -31, -74, 35, 13, -16, 1, 5, -2, -1, 1]);
 

\( x^{44} - x^{43} - 2 x^{42} + 5 x^{41} + x^{40} - 16 x^{39} + 13 x^{38} + 35 x^{37} - 74 x^{36} - 31 x^{35} + 253 x^{34} - 160 x^{33} - 599 x^{32} + 1079 x^{31} + 718 x^{30} - 3955 x^{29} + 1801 x^{28} + 10064 x^{27} - 15467 x^{26} - 14725 x^{25} + 61126 x^{24} - 16951 x^{23} - 166427 x^{22} - 50853 x^{21} + 550134 x^{20} - 397575 x^{19} - 1252827 x^{18} + 2445552 x^{17} + 1312929 x^{16} - 8649585 x^{15} + 4710798 x^{14} + 21237957 x^{13} - 35370351 x^{12} - 28343520 x^{11} + 134454573 x^{10} - 49424013 x^{9} - 353939706 x^{8} + 502211745 x^{7} + 559607373 x^{6} - 2066242608 x^{5} + 387420489 x^{4} + 5811307335 x^{3} - 6973568802 x^{2} - 10460353203 x + 31381059609 \)

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:  \(126\!\cdots\!409\)\(\medspace = 11^{22}\cdot 23^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $66.15$
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:  $11, 23$
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:  \(253=11\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{253}(1,·)$, $\chi_{253}(131,·)$, $\chi_{253}(133,·)$, $\chi_{253}(10,·)$, $\chi_{253}(12,·)$, $\chi_{253}(142,·)$, $\chi_{253}(144,·)$, $\chi_{253}(21,·)$, $\chi_{253}(153,·)$, $\chi_{253}(155,·)$, $\chi_{253}(32,·)$, $\chi_{253}(34,·)$, $\chi_{253}(164,·)$, $\chi_{253}(166,·)$, $\chi_{253}(43,·)$, $\chi_{253}(45,·)$, $\chi_{253}(175,·)$, $\chi_{253}(177,·)$, $\chi_{253}(54,·)$, $\chi_{253}(56,·)$, $\chi_{253}(186,·)$, $\chi_{253}(188,·)$, $\chi_{253}(65,·)$, $\chi_{253}(67,·)$, $\chi_{253}(197,·)$, $\chi_{253}(199,·)$, $\chi_{253}(76,·)$, $\chi_{253}(78,·)$, $\chi_{253}(208,·)$, $\chi_{253}(210,·)$, $\chi_{253}(87,·)$, $\chi_{253}(89,·)$, $\chi_{253}(219,·)$, $\chi_{253}(221,·)$, $\chi_{253}(98,·)$, $\chi_{253}(100,·)$, $\chi_{253}(232,·)$, $\chi_{253}(109,·)$, $\chi_{253}(111,·)$, $\chi_{253}(241,·)$, $\chi_{253}(243,·)$, $\chi_{253}(120,·)$, $\chi_{253}(122,·)$, $\chi_{253}(252,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{499281} a^{23} + \frac{1}{3} a^{22} - \frac{1}{3} a^{21} + \frac{1}{3} a^{20} - \frac{1}{3} a^{19} + \frac{1}{3} a^{18} - \frac{1}{3} a^{17} + \frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{16951}{166427}$, $\frac{1}{1497843} a^{24} - \frac{1}{1497843} a^{23} + \frac{2}{9} a^{22} + \frac{4}{9} a^{21} - \frac{1}{9} a^{20} - \frac{2}{9} a^{19} - \frac{4}{9} a^{18} + \frac{1}{9} a^{17} + \frac{2}{9} a^{16} + \frac{4}{9} a^{15} - \frac{1}{9} a^{14} - \frac{2}{9} a^{13} - \frac{4}{9} a^{12} + \frac{1}{9} a^{11} + \frac{2}{9} a^{10} + \frac{4}{9} a^{9} - \frac{1}{9} a^{8} - \frac{2}{9} a^{7} - \frac{4}{9} a^{6} + \frac{1}{9} a^{5} + \frac{2}{9} a^{4} + \frac{4}{9} a^{3} - \frac{1}{9} a^{2} - \frac{16951}{499281} a + \frac{61126}{166427}$, $\frac{1}{4493529} a^{25} - \frac{1}{4493529} a^{24} - \frac{2}{4493529} a^{23} - \frac{5}{27} a^{22} - \frac{1}{27} a^{21} - \frac{11}{27} a^{20} - \frac{13}{27} a^{19} - \frac{8}{27} a^{18} - \frac{7}{27} a^{17} + \frac{4}{27} a^{16} - \frac{10}{27} a^{15} - \frac{2}{27} a^{14} + \frac{5}{27} a^{13} + \frac{1}{27} a^{12} + \frac{11}{27} a^{11} + \frac{13}{27} a^{10} + \frac{8}{27} a^{9} + \frac{7}{27} a^{8} - \frac{4}{27} a^{7} + \frac{10}{27} a^{6} + \frac{2}{27} a^{5} - \frac{5}{27} a^{4} - \frac{1}{27} a^{3} - \frac{16951}{1497843} a^{2} + \frac{61126}{499281} a - \frac{14725}{166427}$, $\frac{1}{13480587} a^{26} - \frac{1}{13480587} a^{25} - \frac{2}{13480587} a^{24} + \frac{5}{13480587} a^{23} + \frac{26}{81} a^{22} - \frac{11}{81} a^{21} + \frac{14}{81} a^{20} + \frac{19}{81} a^{19} + \frac{20}{81} a^{18} + \frac{4}{81} a^{17} + \frac{17}{81} a^{16} - \frac{29}{81} a^{15} - \frac{22}{81} a^{14} + \frac{28}{81} a^{13} + \frac{38}{81} a^{12} + \frac{40}{81} a^{11} + \frac{8}{81} a^{10} + \frac{34}{81} a^{9} + \frac{23}{81} a^{8} + \frac{37}{81} a^{7} - \frac{25}{81} a^{6} - \frac{5}{81} a^{5} - \frac{1}{81} a^{4} - \frac{16951}{4493529} a^{3} + \frac{61126}{1497843} a^{2} - \frac{14725}{499281} a - \frac{15467}{166427}$, $\frac{1}{40441761} a^{27} - \frac{1}{40441761} a^{26} - \frac{2}{40441761} a^{25} + \frac{5}{40441761} a^{24} + \frac{1}{40441761} a^{23} + \frac{70}{243} a^{22} + \frac{95}{243} a^{21} - \frac{62}{243} a^{20} + \frac{20}{243} a^{19} - \frac{77}{243} a^{18} + \frac{17}{243} a^{17} - \frac{29}{243} a^{16} - \frac{22}{243} a^{15} + \frac{109}{243} a^{14} - \frac{43}{243} a^{13} - \frac{41}{243} a^{12} - \frac{73}{243} a^{11} - \frac{47}{243} a^{10} + \frac{23}{243} a^{9} + \frac{118}{243} a^{8} + \frac{56}{243} a^{7} + \frac{76}{243} a^{6} - \frac{1}{243} a^{5} - \frac{16951}{13480587} a^{4} + \frac{61126}{4493529} a^{3} - \frac{14725}{1497843} a^{2} - \frac{15467}{499281} a + \frac{10064}{166427}$, $\frac{1}{121325283} a^{28} - \frac{1}{121325283} a^{27} - \frac{2}{121325283} a^{26} + \frac{5}{121325283} a^{25} + \frac{1}{121325283} a^{24} - \frac{16}{121325283} a^{23} + \frac{95}{729} a^{22} - \frac{305}{729} a^{21} + \frac{20}{729} a^{20} + \frac{166}{729} a^{19} - \frac{226}{729} a^{18} - \frac{272}{729} a^{17} + \frac{221}{729} a^{16} - \frac{134}{729} a^{15} + \frac{200}{729} a^{14} + \frac{202}{729} a^{13} - \frac{73}{729} a^{12} + \frac{196}{729} a^{11} + \frac{23}{729} a^{10} + \frac{118}{729} a^{9} - \frac{187}{729} a^{8} - \frac{167}{729} a^{7} - \frac{1}{729} a^{6} - \frac{16951}{40441761} a^{5} + \frac{61126}{13480587} a^{4} - \frac{14725}{4493529} a^{3} - \frac{15467}{1497843} a^{2} + \frac{10064}{499281} a + \frac{1801}{166427}$, $\frac{1}{363975849} a^{29} - \frac{1}{363975849} a^{28} - \frac{2}{363975849} a^{27} + \frac{5}{363975849} a^{26} + \frac{1}{363975849} a^{25} - \frac{16}{363975849} a^{24} + \frac{13}{363975849} a^{23} - \frac{305}{2187} a^{22} + \frac{20}{2187} a^{21} + \frac{895}{2187} a^{20} - \frac{955}{2187} a^{19} + \frac{457}{2187} a^{18} + \frac{221}{2187} a^{17} + \frac{595}{2187} a^{16} + \frac{929}{2187} a^{15} - \frac{527}{2187} a^{14} - \frac{73}{2187} a^{13} - \frac{533}{2187} a^{12} + \frac{752}{2187} a^{11} + \frac{847}{2187} a^{10} - \frac{916}{2187} a^{9} + \frac{562}{2187} a^{8} - \frac{1}{2187} a^{7} - \frac{16951}{121325283} a^{6} + \frac{61126}{40441761} a^{5} - \frac{14725}{13480587} a^{4} - \frac{15467}{4493529} a^{3} + \frac{10064}{1497843} a^{2} + \frac{1801}{499281} a - \frac{3955}{166427}$, $\frac{1}{1091927547} a^{30} - \frac{1}{1091927547} a^{29} - \frac{2}{1091927547} a^{28} + \frac{5}{1091927547} a^{27} + \frac{1}{1091927547} a^{26} - \frac{16}{1091927547} a^{25} + \frac{13}{1091927547} a^{24} + \frac{35}{1091927547} a^{23} + \frac{2207}{6561} a^{22} - \frac{1292}{6561} a^{21} + \frac{1232}{6561} a^{20} + \frac{2644}{6561} a^{19} + \frac{221}{6561} a^{18} - \frac{1592}{6561} a^{17} + \frac{929}{6561} a^{16} - \frac{2714}{6561} a^{15} - \frac{73}{6561} a^{14} + \frac{1654}{6561} a^{13} - \frac{1435}{6561} a^{12} + \frac{3034}{6561} a^{11} + \frac{1271}{6561} a^{10} + \frac{2749}{6561} a^{9} - \frac{1}{6561} a^{8} - \frac{16951}{363975849} a^{7} + \frac{61126}{121325283} a^{6} - \frac{14725}{40441761} a^{5} - \frac{15467}{13480587} a^{4} + \frac{10064}{4493529} a^{3} + \frac{1801}{1497843} a^{2} - \frac{3955}{499281} a + \frac{718}{166427}$, $\frac{1}{3275782641} a^{31} - \frac{1}{3275782641} a^{30} - \frac{2}{3275782641} a^{29} + \frac{5}{3275782641} a^{28} + \frac{1}{3275782641} a^{27} - \frac{16}{3275782641} a^{26} + \frac{13}{3275782641} a^{25} + \frac{35}{3275782641} a^{24} - \frac{74}{3275782641} a^{23} - \frac{7853}{19683} a^{22} + \frac{1232}{19683} a^{21} + \frac{2644}{19683} a^{20} - \frac{6340}{19683} a^{19} - \frac{1592}{19683} a^{18} + \frac{929}{19683} a^{17} + \frac{3847}{19683} a^{16} - \frac{6634}{19683} a^{15} - \frac{4907}{19683} a^{14} + \frac{5126}{19683} a^{13} + \frac{9595}{19683} a^{12} - \frac{5290}{19683} a^{11} - \frac{3812}{19683} a^{10} - \frac{1}{19683} a^{9} - \frac{16951}{1091927547} a^{8} + \frac{61126}{363975849} a^{7} - \frac{14725}{121325283} a^{6} - \frac{15467}{40441761} a^{5} + \frac{10064}{13480587} a^{4} + \frac{1801}{4493529} a^{3} - \frac{3955}{1497843} a^{2} + \frac{718}{499281} a + \frac{1079}{166427}$, $\frac{1}{9827347923} a^{32} - \frac{1}{9827347923} a^{31} - \frac{2}{9827347923} a^{30} + \frac{5}{9827347923} a^{29} + \frac{1}{9827347923} a^{28} - \frac{16}{9827347923} a^{27} + \frac{13}{9827347923} a^{26} + \frac{35}{9827347923} a^{25} - \frac{74}{9827347923} a^{24} - \frac{31}{9827347923} a^{23} + \frac{20915}{59049} a^{22} + \frac{2644}{59049} a^{21} - \frac{6340}{59049} a^{20} - \frac{1592}{59049} a^{19} + \frac{20612}{59049} a^{18} - \frac{15836}{59049} a^{17} + \frac{13049}{59049} a^{16} - \frac{24590}{59049} a^{15} - \frac{14557}{59049} a^{14} + \frac{29278}{59049} a^{13} + \frac{14393}{59049} a^{12} + \frac{15871}{59049} a^{11} - \frac{1}{59049} a^{10} - \frac{16951}{3275782641} a^{9} + \frac{61126}{1091927547} a^{8} - \frac{14725}{363975849} a^{7} - \frac{15467}{121325283} a^{6} + \frac{10064}{40441761} a^{5} + \frac{1801}{13480587} a^{4} - \frac{3955}{4493529} a^{3} + \frac{718}{1497843} a^{2} + \frac{1079}{499281} a - \frac{599}{166427}$, $\frac{1}{29482043769} a^{33} - \frac{1}{29482043769} a^{32} - \frac{2}{29482043769} a^{31} + \frac{5}{29482043769} a^{30} + \frac{1}{29482043769} a^{29} - \frac{16}{29482043769} a^{28} + \frac{13}{29482043769} a^{27} + \frac{35}{29482043769} a^{26} - \frac{74}{29482043769} a^{25} - \frac{31}{29482043769} a^{24} + \frac{253}{29482043769} a^{23} + \frac{2644}{177147} a^{22} - \frac{65389}{177147} a^{21} + \frac{57457}{177147} a^{20} - \frac{38437}{177147} a^{19} + \frac{43213}{177147} a^{18} + \frac{72098}{177147} a^{17} - \frac{24590}{177147} a^{16} - \frac{14557}{177147} a^{15} + \frac{88327}{177147} a^{14} - \frac{44656}{177147} a^{13} - \frac{43178}{177147} a^{12} - \frac{1}{177147} a^{11} - \frac{16951}{9827347923} a^{10} + \frac{61126}{3275782641} a^{9} - \frac{14725}{1091927547} a^{8} - \frac{15467}{363975849} a^{7} + \frac{10064}{121325283} a^{6} + \frac{1801}{40441761} a^{5} - \frac{3955}{13480587} a^{4} + \frac{718}{4493529} a^{3} + \frac{1079}{1497843} a^{2} - \frac{599}{499281} a - \frac{160}{166427}$, $\frac{1}{88446131307} a^{34} - \frac{1}{88446131307} a^{33} - \frac{2}{88446131307} a^{32} + \frac{5}{88446131307} a^{31} + \frac{1}{88446131307} a^{30} - \frac{16}{88446131307} a^{29} + \frac{13}{88446131307} a^{28} + \frac{35}{88446131307} a^{27} - \frac{74}{88446131307} a^{26} - \frac{31}{88446131307} a^{25} + \frac{253}{88446131307} a^{24} - \frac{160}{88446131307} a^{23} - \frac{242536}{531441} a^{22} + \frac{234604}{531441} a^{21} - \frac{38437}{531441} a^{20} - \frac{133934}{531441} a^{19} + \frac{249245}{531441} a^{18} + \frac{152557}{531441} a^{17} + \frac{162590}{531441} a^{16} - \frac{88820}{531441} a^{15} + \frac{132491}{531441} a^{14} + \frac{133969}{531441} a^{13} - \frac{1}{531441} a^{12} - \frac{16951}{29482043769} a^{11} + \frac{61126}{9827347923} a^{10} - \frac{14725}{3275782641} a^{9} - \frac{15467}{1091927547} a^{8} + \frac{10064}{363975849} a^{7} + \frac{1801}{121325283} a^{6} - \frac{3955}{40441761} a^{5} + \frac{718}{13480587} a^{4} + \frac{1079}{4493529} a^{3} - \frac{599}{1497843} a^{2} - \frac{160}{499281} a + \frac{253}{166427}$, $\frac{1}{265338393921} a^{35} - \frac{1}{265338393921} a^{34} - \frac{2}{265338393921} a^{33} + \frac{5}{265338393921} a^{32} + \frac{1}{265338393921} a^{31} - \frac{16}{265338393921} a^{30} + \frac{13}{265338393921} a^{29} + \frac{35}{265338393921} a^{28} - \frac{74}{265338393921} a^{27} - \frac{31}{265338393921} a^{26} + \frac{253}{265338393921} a^{25} - \frac{160}{265338393921} a^{24} - \frac{599}{265338393921} a^{23} - \frac{296837}{1594323} a^{22} - \frac{569878}{1594323} a^{21} - \frac{133934}{1594323} a^{20} + \frac{249245}{1594323} a^{19} + \frac{152557}{1594323} a^{18} + \frac{694031}{1594323} a^{17} + \frac{442621}{1594323} a^{16} + \frac{663932}{1594323} a^{15} - \frac{397472}{1594323} a^{14} - \frac{1}{1594323} a^{13} - \frac{16951}{88446131307} a^{12} + \frac{61126}{29482043769} a^{11} - \frac{14725}{9827347923} a^{10} - \frac{15467}{3275782641} a^{9} + \frac{10064}{1091927547} a^{8} + \frac{1801}{363975849} a^{7} - \frac{3955}{121325283} a^{6} + \frac{718}{40441761} a^{5} + \frac{1079}{13480587} a^{4} - \frac{599}{4493529} a^{3} - \frac{160}{1497843} a^{2} + \frac{253}{499281} a - \frac{31}{166427}$, $\frac{1}{796015181763} a^{36} - \frac{1}{796015181763} a^{35} - \frac{2}{796015181763} a^{34} + \frac{5}{796015181763} a^{33} + \frac{1}{796015181763} a^{32} - \frac{16}{796015181763} a^{31} + \frac{13}{796015181763} a^{30} + \frac{35}{796015181763} a^{29} - \frac{74}{796015181763} a^{28} - \frac{31}{796015181763} a^{27} + \frac{253}{796015181763} a^{26} - \frac{160}{796015181763} a^{25} - \frac{599}{796015181763} a^{24} + \frac{1079}{796015181763} a^{23} - \frac{2164201}{4782969} a^{22} - \frac{1728257}{4782969} a^{21} - \frac{1345078}{4782969} a^{20} + \frac{1746880}{4782969} a^{19} + \frac{2288354}{4782969} a^{18} + \frac{2036944}{4782969} a^{17} + \frac{663932}{4782969} a^{16} - \frac{1991795}{4782969} a^{15} - \frac{1}{4782969} a^{14} - \frac{16951}{265338393921} a^{13} + \frac{61126}{88446131307} a^{12} - \frac{14725}{29482043769} a^{11} - \frac{15467}{9827347923} a^{10} + \frac{10064}{3275782641} a^{9} + \frac{1801}{1091927547} a^{8} - \frac{3955}{363975849} a^{7} + \frac{718}{121325283} a^{6} + \frac{1079}{40441761} a^{5} - \frac{599}{13480587} a^{4} - \frac{160}{4493529} a^{3} + \frac{253}{1497843} a^{2} - \frac{31}{499281} a - \frac{74}{166427}$, $\frac{1}{2388045545289} a^{37} - \frac{1}{2388045545289} a^{36} - \frac{2}{2388045545289} a^{35} + \frac{5}{2388045545289} a^{34} + \frac{1}{2388045545289} a^{33} - \frac{16}{2388045545289} a^{32} + \frac{13}{2388045545289} a^{31} + \frac{35}{2388045545289} a^{30} - \frac{74}{2388045545289} a^{29} - \frac{31}{2388045545289} a^{28} + \frac{253}{2388045545289} a^{27} - \frac{160}{2388045545289} a^{26} - \frac{599}{2388045545289} a^{25} + \frac{1079}{2388045545289} a^{24} + \frac{718}{2388045545289} a^{23} - \frac{6511226}{14348907} a^{22} - \frac{1345078}{14348907} a^{21} + \frac{6529849}{14348907} a^{20} - \frac{2494615}{14348907} a^{19} - \frac{2746025}{14348907} a^{18} - \frac{4119037}{14348907} a^{17} - \frac{1991795}{14348907} a^{16} - \frac{1}{14348907} a^{15} - \frac{16951}{796015181763} a^{14} + \frac{61126}{265338393921} a^{13} - \frac{14725}{88446131307} a^{12} - \frac{15467}{29482043769} a^{11} + \frac{10064}{9827347923} a^{10} + \frac{1801}{3275782641} a^{9} - \frac{3955}{1091927547} a^{8} + \frac{718}{363975849} a^{7} + \frac{1079}{121325283} a^{6} - \frac{599}{40441761} a^{5} - \frac{160}{13480587} a^{4} + \frac{253}{4493529} a^{3} - \frac{31}{1497843} a^{2} - \frac{74}{499281} a + \frac{35}{166427}$, $\frac{1}{7164136635867} a^{38} - \frac{1}{7164136635867} a^{37} - \frac{2}{7164136635867} a^{36} + \frac{5}{7164136635867} a^{35} + \frac{1}{7164136635867} a^{34} - \frac{16}{7164136635867} a^{33} + \frac{13}{7164136635867} a^{32} + \frac{35}{7164136635867} a^{31} - \frac{74}{7164136635867} a^{30} - \frac{31}{7164136635867} a^{29} + \frac{253}{7164136635867} a^{28} - \frac{160}{7164136635867} a^{27} - \frac{599}{7164136635867} a^{26} + \frac{1079}{7164136635867} a^{25} + \frac{718}{7164136635867} a^{24} - \frac{3955}{7164136635867} a^{23} - \frac{15693985}{43046721} a^{22} - \frac{7819058}{43046721} a^{21} + \frac{11854292}{43046721} a^{20} + \frac{11602882}{43046721} a^{19} - \frac{4119037}{43046721} a^{18} + \frac{12357112}{43046721} a^{17} - \frac{1}{43046721} a^{16} - \frac{16951}{2388045545289} a^{15} + \frac{61126}{796015181763} a^{14} - \frac{14725}{265338393921} a^{13} - \frac{15467}{88446131307} a^{12} + \frac{10064}{29482043769} a^{11} + \frac{1801}{9827347923} a^{10} - \frac{3955}{3275782641} a^{9} + \frac{718}{1091927547} a^{8} + \frac{1079}{363975849} a^{7} - \frac{599}{121325283} a^{6} - \frac{160}{40441761} a^{5} + \frac{253}{13480587} a^{4} - \frac{31}{4493529} a^{3} - \frac{74}{1497843} a^{2} + \frac{35}{499281} a + \frac{13}{166427}$, $\frac{1}{21492409907601} a^{39} - \frac{1}{21492409907601} a^{38} - \frac{2}{21492409907601} a^{37} + \frac{5}{21492409907601} a^{36} + \frac{1}{21492409907601} a^{35} - \frac{16}{21492409907601} a^{34} + \frac{13}{21492409907601} a^{33} + \frac{35}{21492409907601} a^{32} - \frac{74}{21492409907601} a^{31} - \frac{31}{21492409907601} a^{30} + \frac{253}{21492409907601} a^{29} - \frac{160}{21492409907601} a^{28} - \frac{599}{21492409907601} a^{27} + \frac{1079}{21492409907601} a^{26} + \frac{718}{21492409907601} a^{25} - \frac{3955}{21492409907601} a^{24} + \frac{1801}{21492409907601} a^{23} + \frac{35227663}{129140163} a^{22} + \frac{11854292}{129140163} a^{21} + \frac{11602882}{129140163} a^{20} - \frac{47165758}{129140163} a^{19} + \frac{12357112}{129140163} a^{18} - \frac{1}{129140163} a^{17} - \frac{16951}{7164136635867} a^{16} + \frac{61126}{2388045545289} a^{15} - \frac{14725}{796015181763} a^{14} - \frac{15467}{265338393921} a^{13} + \frac{10064}{88446131307} a^{12} + \frac{1801}{29482043769} a^{11} - \frac{3955}{9827347923} a^{10} + \frac{718}{3275782641} a^{9} + \frac{1079}{1091927547} a^{8} - \frac{599}{363975849} a^{7} - \frac{160}{121325283} a^{6} + \frac{253}{40441761} a^{5} - \frac{31}{13480587} a^{4} - \frac{74}{4493529} a^{3} + \frac{35}{1497843} a^{2} + \frac{13}{499281} a - \frac{16}{166427}$, $\frac{1}{64477229722803} a^{40} - \frac{1}{64477229722803} a^{39} - \frac{2}{64477229722803} a^{38} + \frac{5}{64477229722803} a^{37} + \frac{1}{64477229722803} a^{36} - \frac{16}{64477229722803} a^{35} + \frac{13}{64477229722803} a^{34} + \frac{35}{64477229722803} a^{33} - \frac{74}{64477229722803} a^{32} - \frac{31}{64477229722803} a^{31} + \frac{253}{64477229722803} a^{30} - \frac{160}{64477229722803} a^{29} - \frac{599}{64477229722803} a^{28} + \frac{1079}{64477229722803} a^{27} + \frac{718}{64477229722803} a^{26} - \frac{3955}{64477229722803} a^{25} + \frac{1801}{64477229722803} a^{24} + \frac{10064}{64477229722803} a^{23} + \frac{140994455}{387420489} a^{22} + \frac{140743045}{387420489} a^{21} - \frac{176305921}{387420489} a^{20} + \frac{141497275}{387420489} a^{19} - \frac{1}{387420489} a^{18} - \frac{16951}{21492409907601} a^{17} + \frac{61126}{7164136635867} a^{16} - \frac{14725}{2388045545289} a^{15} - \frac{15467}{796015181763} a^{14} + \frac{10064}{265338393921} a^{13} + \frac{1801}{88446131307} a^{12} - \frac{3955}{29482043769} a^{11} + \frac{718}{9827347923} a^{10} + \frac{1079}{3275782641} a^{9} - \frac{599}{1091927547} a^{8} - \frac{160}{363975849} a^{7} + \frac{253}{121325283} a^{6} - \frac{31}{40441761} a^{5} - \frac{74}{13480587} a^{4} + \frac{35}{4493529} a^{3} + \frac{13}{1497843} a^{2} - \frac{16}{499281} a + \frac{1}{166427}$, $\frac{1}{193431689168409} a^{41} - \frac{1}{193431689168409} a^{40} - \frac{2}{193431689168409} a^{39} + \frac{5}{193431689168409} a^{38} + \frac{1}{193431689168409} a^{37} - \frac{16}{193431689168409} a^{36} + \frac{13}{193431689168409} a^{35} + \frac{35}{193431689168409} a^{34} - \frac{74}{193431689168409} a^{33} - \frac{31}{193431689168409} a^{32} + \frac{253}{193431689168409} a^{31} - \frac{160}{193431689168409} a^{30} - \frac{599}{193431689168409} a^{29} + \frac{1079}{193431689168409} a^{28} + \frac{718}{193431689168409} a^{27} - \frac{3955}{193431689168409} a^{26} + \frac{1801}{193431689168409} a^{25} + \frac{10064}{193431689168409} a^{24} - \frac{15467}{193431689168409} a^{23} + \frac{528163534}{1162261467} a^{22} + \frac{211114568}{1162261467} a^{21} + \frac{528917764}{1162261467} a^{20} - \frac{1}{1162261467} a^{19} - \frac{16951}{64477229722803} a^{18} + \frac{61126}{21492409907601} a^{17} - \frac{14725}{7164136635867} a^{16} - \frac{15467}{2388045545289} a^{15} + \frac{10064}{796015181763} a^{14} + \frac{1801}{265338393921} a^{13} - \frac{3955}{88446131307} a^{12} + \frac{718}{29482043769} a^{11} + \frac{1079}{9827347923} a^{10} - \frac{599}{3275782641} a^{9} - \frac{160}{1091927547} a^{8} + \frac{253}{363975849} a^{7} - \frac{31}{121325283} a^{6} - \frac{74}{40441761} a^{5} + \frac{35}{13480587} a^{4} + \frac{13}{4493529} a^{3} - \frac{16}{1497843} a^{2} + \frac{1}{499281} a + \frac{5}{166427}$, $\frac{1}{580295067505227} a^{42} - \frac{1}{580295067505227} a^{41} - \frac{2}{580295067505227} a^{40} + \frac{5}{580295067505227} a^{39} + \frac{1}{580295067505227} a^{38} - \frac{16}{580295067505227} a^{37} + \frac{13}{580295067505227} a^{36} + \frac{35}{580295067505227} a^{35} - \frac{74}{580295067505227} a^{34} - \frac{31}{580295067505227} a^{33} + \frac{253}{580295067505227} a^{32} - \frac{160}{580295067505227} a^{31} - \frac{599}{580295067505227} a^{30} + \frac{1079}{580295067505227} a^{29} + \frac{718}{580295067505227} a^{28} - \frac{3955}{580295067505227} a^{27} + \frac{1801}{580295067505227} a^{26} + \frac{10064}{580295067505227} a^{25} - \frac{15467}{580295067505227} a^{24} - \frac{14725}{580295067505227} a^{23} - \frac{951146899}{3486784401} a^{22} - \frac{633343703}{3486784401} a^{21} - \frac{1}{3486784401} a^{20} - \frac{16951}{193431689168409} a^{19} + \frac{61126}{64477229722803} a^{18} - \frac{14725}{21492409907601} a^{17} - \frac{15467}{7164136635867} a^{16} + \frac{10064}{2388045545289} a^{15} + \frac{1801}{796015181763} a^{14} - \frac{3955}{265338393921} a^{13} + \frac{718}{88446131307} a^{12} + \frac{1079}{29482043769} a^{11} - \frac{599}{9827347923} a^{10} - \frac{160}{3275782641} a^{9} + \frac{253}{1091927547} a^{8} - \frac{31}{363975849} a^{7} - \frac{74}{121325283} a^{6} + \frac{35}{40441761} a^{5} + \frac{13}{13480587} a^{4} - \frac{16}{4493529} a^{3} + \frac{1}{1497843} a^{2} + \frac{5}{499281} a - \frac{2}{166427}$, $\frac{1}{1740885202515681} a^{43} - \frac{1}{1740885202515681} a^{42} - \frac{2}{1740885202515681} a^{41} + \frac{5}{1740885202515681} a^{40} + \frac{1}{1740885202515681} a^{39} - \frac{16}{1740885202515681} a^{38} + \frac{13}{1740885202515681} a^{37} + \frac{35}{1740885202515681} a^{36} - \frac{74}{1740885202515681} a^{35} - \frac{31}{1740885202515681} a^{34} + \frac{253}{1740885202515681} a^{33} - \frac{160}{1740885202515681} a^{32} - \frac{599}{1740885202515681} a^{31} + \frac{1079}{1740885202515681} a^{30} + \frac{718}{1740885202515681} a^{29} - \frac{3955}{1740885202515681} a^{28} + \frac{1801}{1740885202515681} a^{27} + \frac{10064}{1740885202515681} a^{26} - \frac{15467}{1740885202515681} a^{25} - \frac{14725}{1740885202515681} a^{24} + \frac{61126}{1740885202515681} a^{23} + \frac{2853440698}{10460353203} a^{22} - \frac{1}{10460353203} a^{21} - \frac{16951}{580295067505227} a^{20} + \frac{61126}{193431689168409} a^{19} - \frac{14725}{64477229722803} a^{18} - \frac{15467}{21492409907601} a^{17} + \frac{10064}{7164136635867} a^{16} + \frac{1801}{2388045545289} a^{15} - \frac{3955}{796015181763} a^{14} + \frac{718}{265338393921} a^{13} + \frac{1079}{88446131307} a^{12} - \frac{599}{29482043769} a^{11} - \frac{160}{9827347923} a^{10} + \frac{253}{3275782641} a^{9} - \frac{31}{1091927547} a^{8} - \frac{74}{363975849} a^{7} + \frac{35}{121325283} a^{6} + \frac{13}{40441761} a^{5} - \frac{16}{13480587} a^{4} + \frac{1}{4493529} a^{3} + \frac{5}{1497843} a^{2} - \frac{2}{499281} a - \frac{1}{166427}$

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:  \( \frac{15467}{193431689168409} a^{42} + \frac{1664094224}{193431689168409} a^{19} \) (order $46$)
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

$C_2\times C_{22}$ (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

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{253}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-11}, \sqrt{-23})\), \(\Q(\zeta_{23})^+\), 22.0.489639287266880371204241478228795637811.1, 22.22.11261703607138248537697553999262299669653.1, \(\Q(\zeta_{23})\)

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 $22^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ R $22^{2}$ $22^{2}$ $22^{2}$ R $22^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{44}$ $22^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{4}$

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
11Data not computed
23Data not computed