Properties

Label 44.0.609...321.1
Degree $44$
Signature $[0, 22]$
Discriminant $6.092\times 10^{75}$
Root discriminant $52.77$
Ramified primes $7, 23$
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 - x^42 + 3*x^41 - x^40 - 5*x^39 + 7*x^38 + 3*x^37 - 17*x^36 + 11*x^35 + 23*x^34 - 45*x^33 - x^32 + 91*x^31 - 89*x^30 - 93*x^29 + 271*x^28 - 85*x^27 - 457*x^26 + 627*x^25 + 287*x^24 - 1541*x^23 + 967*x^22 - 3082*x^21 + 1148*x^20 + 5016*x^19 - 7312*x^18 - 2720*x^17 + 17344*x^16 - 11904*x^15 - 22784*x^14 + 46592*x^13 - 1024*x^12 - 92160*x^11 + 94208*x^10 + 90112*x^9 - 278528*x^8 + 98304*x^7 + 458752*x^6 - 655360*x^5 - 262144*x^4 + 1572864*x^3 - 1048576*x^2 - 2097152*x + 4194304)
 
gp: K = bnfinit(x^44 - x^43 - x^42 + 3*x^41 - x^40 - 5*x^39 + 7*x^38 + 3*x^37 - 17*x^36 + 11*x^35 + 23*x^34 - 45*x^33 - x^32 + 91*x^31 - 89*x^30 - 93*x^29 + 271*x^28 - 85*x^27 - 457*x^26 + 627*x^25 + 287*x^24 - 1541*x^23 + 967*x^22 - 3082*x^21 + 1148*x^20 + 5016*x^19 - 7312*x^18 - 2720*x^17 + 17344*x^16 - 11904*x^15 - 22784*x^14 + 46592*x^13 - 1024*x^12 - 92160*x^11 + 94208*x^10 + 90112*x^9 - 278528*x^8 + 98304*x^7 + 458752*x^6 - 655360*x^5 - 262144*x^4 + 1572864*x^3 - 1048576*x^2 - 2097152*x + 4194304, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4194304, -2097152, -1048576, 1572864, -262144, -655360, 458752, 98304, -278528, 90112, 94208, -92160, -1024, 46592, -22784, -11904, 17344, -2720, -7312, 5016, 1148, -3082, 967, -1541, 287, 627, -457, -85, 271, -93, -89, 91, -1, -45, 23, 11, -17, 3, 7, -5, -1, 3, -1, -1, 1]);
 

\( x^{44} - x^{43} - x^{42} + 3 x^{41} - x^{40} - 5 x^{39} + 7 x^{38} + 3 x^{37} - 17 x^{36} + 11 x^{35} + 23 x^{34} - 45 x^{33} - x^{32} + 91 x^{31} - 89 x^{30} - 93 x^{29} + 271 x^{28} - 85 x^{27} - 457 x^{26} + 627 x^{25} + 287 x^{24} - 1541 x^{23} + 967 x^{22} - 3082 x^{21} + 1148 x^{20} + 5016 x^{19} - 7312 x^{18} - 2720 x^{17} + 17344 x^{16} - 11904 x^{15} - 22784 x^{14} + 46592 x^{13} - 1024 x^{12} - 92160 x^{11} + 94208 x^{10} + 90112 x^{9} - 278528 x^{8} + 98304 x^{7} + 458752 x^{6} - 655360 x^{5} - 262144 x^{4} + 1572864 x^{3} - 1048576 x^{2} - 2097152 x + 4194304 \)

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:  \(609\!\cdots\!321\)\(\medspace = 7^{22}\cdot 23^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $52.77$
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:  $7, 23$
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:  \(161=7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{161}(1,·)$, $\chi_{161}(132,·)$, $\chi_{161}(134,·)$, $\chi_{161}(8,·)$, $\chi_{161}(139,·)$, $\chi_{161}(13,·)$, $\chi_{161}(15,·)$, $\chi_{161}(146,·)$, $\chi_{161}(20,·)$, $\chi_{161}(22,·)$, $\chi_{161}(153,·)$, $\chi_{161}(155,·)$, $\chi_{161}(29,·)$, $\chi_{161}(160,·)$, $\chi_{161}(34,·)$, $\chi_{161}(27,·)$, $\chi_{161}(36,·)$, $\chi_{161}(6,·)$, $\chi_{161}(41,·)$, $\chi_{161}(43,·)$, $\chi_{161}(48,·)$, $\chi_{161}(50,·)$, $\chi_{161}(55,·)$, $\chi_{161}(57,·)$, $\chi_{161}(62,·)$, $\chi_{161}(64,·)$, $\chi_{161}(71,·)$, $\chi_{161}(76,·)$, $\chi_{161}(78,·)$, $\chi_{161}(141,·)$, $\chi_{161}(83,·)$, $\chi_{161}(85,·)$, $\chi_{161}(90,·)$, $\chi_{161}(97,·)$, $\chi_{161}(99,·)$, $\chi_{161}(104,·)$, $\chi_{161}(106,·)$, $\chi_{161}(111,·)$, $\chi_{161}(113,·)$, $\chi_{161}(118,·)$, $\chi_{161}(120,·)$, $\chi_{161}(148,·)$, $\chi_{161}(125,·)$, $\chi_{161}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{1934} a^{23} - \frac{1}{2} a^{22} - \frac{1}{2} a^{21} - \frac{1}{2} a^{20} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{393}{967}$, $\frac{1}{3868} a^{24} - \frac{1}{3868} a^{23} + \frac{1}{4} a^{22} + \frac{1}{4} a^{21} + \frac{1}{4} a^{20} + \frac{1}{4} a^{19} + \frac{1}{4} a^{18} + \frac{1}{4} a^{17} + \frac{1}{4} a^{16} + \frac{1}{4} a^{15} + \frac{1}{4} a^{14} + \frac{1}{4} a^{13} + \frac{1}{4} a^{12} + \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{393}{1934} a + \frac{287}{967}$, $\frac{1}{7736} a^{25} - \frac{1}{7736} a^{24} - \frac{1}{7736} a^{23} - \frac{3}{8} a^{22} + \frac{1}{8} a^{21} - \frac{3}{8} a^{20} + \frac{1}{8} a^{19} - \frac{3}{8} a^{18} + \frac{1}{8} a^{17} - \frac{3}{8} a^{16} + \frac{1}{8} a^{15} - \frac{3}{8} a^{14} + \frac{1}{8} a^{13} - \frac{3}{8} a^{12} + \frac{1}{8} a^{11} - \frac{3}{8} a^{10} + \frac{1}{8} a^{9} - \frac{3}{8} a^{8} + \frac{1}{8} a^{7} - \frac{3}{8} a^{6} + \frac{1}{8} a^{5} - \frac{3}{8} a^{4} + \frac{1}{8} a^{3} - \frac{1541}{3868} a^{2} + \frac{287}{1934} a - \frac{340}{967}$, $\frac{1}{15472} a^{26} - \frac{1}{15472} a^{25} - \frac{1}{15472} a^{24} + \frac{3}{15472} a^{23} - \frac{7}{16} a^{22} - \frac{3}{16} a^{21} + \frac{1}{16} a^{20} + \frac{5}{16} a^{19} - \frac{7}{16} a^{18} - \frac{3}{16} a^{17} + \frac{1}{16} a^{16} + \frac{5}{16} a^{15} - \frac{7}{16} a^{14} - \frac{3}{16} a^{13} + \frac{1}{16} a^{12} + \frac{5}{16} a^{11} - \frac{7}{16} a^{10} - \frac{3}{16} a^{9} + \frac{1}{16} a^{8} + \frac{5}{16} a^{7} - \frac{7}{16} a^{6} - \frac{3}{16} a^{5} + \frac{1}{16} a^{4} - \frac{1541}{7736} a^{3} + \frac{287}{3868} a^{2} + \frac{627}{1934} a - \frac{457}{967}$, $\frac{1}{30944} a^{27} - \frac{1}{30944} a^{26} - \frac{1}{30944} a^{25} + \frac{3}{30944} a^{24} - \frac{1}{30944} a^{23} + \frac{13}{32} a^{22} + \frac{1}{32} a^{21} + \frac{5}{32} a^{20} - \frac{7}{32} a^{19} - \frac{3}{32} a^{18} - \frac{15}{32} a^{17} - \frac{11}{32} a^{16} + \frac{9}{32} a^{15} + \frac{13}{32} a^{14} + \frac{1}{32} a^{13} + \frac{5}{32} a^{12} - \frac{7}{32} a^{11} - \frac{3}{32} a^{10} - \frac{15}{32} a^{9} - \frac{11}{32} a^{8} + \frac{9}{32} a^{7} + \frac{13}{32} a^{6} + \frac{1}{32} a^{5} - \frac{1541}{15472} a^{4} + \frac{287}{7736} a^{3} + \frac{627}{3868} a^{2} - \frac{457}{1934} a - \frac{85}{967}$, $\frac{1}{61888} a^{28} - \frac{1}{61888} a^{27} - \frac{1}{61888} a^{26} + \frac{3}{61888} a^{25} - \frac{1}{61888} a^{24} - \frac{5}{61888} a^{23} + \frac{1}{64} a^{22} - \frac{27}{64} a^{21} + \frac{25}{64} a^{20} + \frac{29}{64} a^{19} - \frac{15}{64} a^{18} + \frac{21}{64} a^{17} + \frac{9}{64} a^{16} + \frac{13}{64} a^{15} - \frac{31}{64} a^{14} + \frac{5}{64} a^{13} - \frac{7}{64} a^{12} - \frac{3}{64} a^{11} + \frac{17}{64} a^{10} - \frac{11}{64} a^{9} - \frac{23}{64} a^{8} - \frac{19}{64} a^{7} + \frac{1}{64} a^{6} - \frac{1541}{30944} a^{5} + \frac{287}{15472} a^{4} + \frac{627}{7736} a^{3} - \frac{457}{3868} a^{2} - \frac{85}{1934} a + \frac{271}{967}$, $\frac{1}{123776} a^{29} - \frac{1}{123776} a^{28} - \frac{1}{123776} a^{27} + \frac{3}{123776} a^{26} - \frac{1}{123776} a^{25} - \frac{5}{123776} a^{24} + \frac{7}{123776} a^{23} - \frac{27}{128} a^{22} + \frac{25}{128} a^{21} + \frac{29}{128} a^{20} + \frac{49}{128} a^{19} + \frac{21}{128} a^{18} + \frac{9}{128} a^{17} - \frac{51}{128} a^{16} + \frac{33}{128} a^{15} - \frac{59}{128} a^{14} - \frac{7}{128} a^{13} - \frac{3}{128} a^{12} + \frac{17}{128} a^{11} - \frac{11}{128} a^{10} - \frac{23}{128} a^{9} + \frac{45}{128} a^{8} + \frac{1}{128} a^{7} - \frac{1541}{61888} a^{6} + \frac{287}{30944} a^{5} + \frac{627}{15472} a^{4} - \frac{457}{7736} a^{3} - \frac{85}{3868} a^{2} + \frac{271}{1934} a - \frac{93}{967}$, $\frac{1}{247552} a^{30} - \frac{1}{247552} a^{29} - \frac{1}{247552} a^{28} + \frac{3}{247552} a^{27} - \frac{1}{247552} a^{26} - \frac{5}{247552} a^{25} + \frac{7}{247552} a^{24} + \frac{3}{247552} a^{23} - \frac{103}{256} a^{22} - \frac{99}{256} a^{21} + \frac{49}{256} a^{20} - \frac{107}{256} a^{19} + \frac{9}{256} a^{18} - \frac{51}{256} a^{17} + \frac{33}{256} a^{16} + \frac{69}{256} a^{15} + \frac{121}{256} a^{14} - \frac{3}{256} a^{13} + \frac{17}{256} a^{12} - \frac{11}{256} a^{11} - \frac{23}{256} a^{10} + \frac{45}{256} a^{9} + \frac{1}{256} a^{8} - \frac{1541}{123776} a^{7} + \frac{287}{61888} a^{6} + \frac{627}{30944} a^{5} - \frac{457}{15472} a^{4} - \frac{85}{7736} a^{3} + \frac{271}{3868} a^{2} - \frac{93}{1934} a - \frac{89}{967}$, $\frac{1}{495104} a^{31} - \frac{1}{495104} a^{30} - \frac{1}{495104} a^{29} + \frac{3}{495104} a^{28} - \frac{1}{495104} a^{27} - \frac{5}{495104} a^{26} + \frac{7}{495104} a^{25} + \frac{3}{495104} a^{24} - \frac{17}{495104} a^{23} - \frac{99}{512} a^{22} - \frac{207}{512} a^{21} - \frac{107}{512} a^{20} + \frac{9}{512} a^{19} + \frac{205}{512} a^{18} - \frac{223}{512} a^{17} - \frac{187}{512} a^{16} + \frac{121}{512} a^{15} + \frac{253}{512} a^{14} + \frac{17}{512} a^{13} - \frac{11}{512} a^{12} - \frac{23}{512} a^{11} + \frac{45}{512} a^{10} + \frac{1}{512} a^{9} - \frac{1541}{247552} a^{8} + \frac{287}{123776} a^{7} + \frac{627}{61888} a^{6} - \frac{457}{30944} a^{5} - \frac{85}{15472} a^{4} + \frac{271}{7736} a^{3} - \frac{93}{3868} a^{2} - \frac{89}{1934} a + \frac{91}{967}$, $\frac{1}{990208} a^{32} - \frac{1}{990208} a^{31} - \frac{1}{990208} a^{30} + \frac{3}{990208} a^{29} - \frac{1}{990208} a^{28} - \frac{5}{990208} a^{27} + \frac{7}{990208} a^{26} + \frac{3}{990208} a^{25} - \frac{17}{990208} a^{24} + \frac{11}{990208} a^{23} + \frac{305}{1024} a^{22} - \frac{107}{1024} a^{21} - \frac{503}{1024} a^{20} - \frac{307}{1024} a^{19} + \frac{289}{1024} a^{18} + \frac{325}{1024} a^{17} + \frac{121}{1024} a^{16} + \frac{253}{1024} a^{15} - \frac{495}{1024} a^{14} - \frac{11}{1024} a^{13} - \frac{23}{1024} a^{12} + \frac{45}{1024} a^{11} + \frac{1}{1024} a^{10} - \frac{1541}{495104} a^{9} + \frac{287}{247552} a^{8} + \frac{627}{123776} a^{7} - \frac{457}{61888} a^{6} - \frac{85}{30944} a^{5} + \frac{271}{15472} a^{4} - \frac{93}{7736} a^{3} - \frac{89}{3868} a^{2} + \frac{91}{1934} a - \frac{1}{967}$, $\frac{1}{1980416} a^{33} - \frac{1}{1980416} a^{32} - \frac{1}{1980416} a^{31} + \frac{3}{1980416} a^{30} - \frac{1}{1980416} a^{29} - \frac{5}{1980416} a^{28} + \frac{7}{1980416} a^{27} + \frac{3}{1980416} a^{26} - \frac{17}{1980416} a^{25} + \frac{11}{1980416} a^{24} + \frac{23}{1980416} a^{23} + \frac{917}{2048} a^{22} + \frac{521}{2048} a^{21} - \frac{307}{2048} a^{20} - \frac{735}{2048} a^{19} - \frac{699}{2048} a^{18} + \frac{121}{2048} a^{17} - \frac{771}{2048} a^{16} + \frac{529}{2048} a^{15} + \frac{1013}{2048} a^{14} - \frac{23}{2048} a^{13} + \frac{45}{2048} a^{12} + \frac{1}{2048} a^{11} - \frac{1541}{990208} a^{10} + \frac{287}{495104} a^{9} + \frac{627}{247552} a^{8} - \frac{457}{123776} a^{7} - \frac{85}{61888} a^{6} + \frac{271}{30944} a^{5} - \frac{93}{15472} a^{4} - \frac{89}{7736} a^{3} + \frac{91}{3868} a^{2} - \frac{1}{1934} a - \frac{45}{967}$, $\frac{1}{3960832} a^{34} - \frac{1}{3960832} a^{33} - \frac{1}{3960832} a^{32} + \frac{3}{3960832} a^{31} - \frac{1}{3960832} a^{30} - \frac{5}{3960832} a^{29} + \frac{7}{3960832} a^{28} + \frac{3}{3960832} a^{27} - \frac{17}{3960832} a^{26} + \frac{11}{3960832} a^{25} + \frac{23}{3960832} a^{24} - \frac{45}{3960832} a^{23} + \frac{521}{4096} a^{22} + \frac{1741}{4096} a^{21} + \frac{1313}{4096} a^{20} - \frac{699}{4096} a^{19} - \frac{1927}{4096} a^{18} - \frac{771}{4096} a^{17} + \frac{529}{4096} a^{16} + \frac{1013}{4096} a^{15} + \frac{2025}{4096} a^{14} + \frac{45}{4096} a^{13} + \frac{1}{4096} a^{12} - \frac{1541}{1980416} a^{11} + \frac{287}{990208} a^{10} + \frac{627}{495104} a^{9} - \frac{457}{247552} a^{8} - \frac{85}{123776} a^{7} + \frac{271}{61888} a^{6} - \frac{93}{30944} a^{5} - \frac{89}{15472} a^{4} + \frac{91}{7736} a^{3} - \frac{1}{3868} a^{2} - \frac{45}{1934} a + \frac{23}{967}$, $\frac{1}{7921664} a^{35} - \frac{1}{7921664} a^{34} - \frac{1}{7921664} a^{33} + \frac{3}{7921664} a^{32} - \frac{1}{7921664} a^{31} - \frac{5}{7921664} a^{30} + \frac{7}{7921664} a^{29} + \frac{3}{7921664} a^{28} - \frac{17}{7921664} a^{27} + \frac{11}{7921664} a^{26} + \frac{23}{7921664} a^{25} - \frac{45}{7921664} a^{24} - \frac{1}{7921664} a^{23} - \frac{2355}{8192} a^{22} + \frac{1313}{8192} a^{21} + \frac{3397}{8192} a^{20} + \frac{2169}{8192} a^{19} - \frac{771}{8192} a^{18} - \frac{3567}{8192} a^{17} - \frac{3083}{8192} a^{16} + \frac{2025}{8192} a^{15} - \frac{4051}{8192} a^{14} + \frac{1}{8192} a^{13} - \frac{1541}{3960832} a^{12} + \frac{287}{1980416} a^{11} + \frac{627}{990208} a^{10} - \frac{457}{495104} a^{9} - \frac{85}{247552} a^{8} + \frac{271}{123776} a^{7} - \frac{93}{61888} a^{6} - \frac{89}{30944} a^{5} + \frac{91}{15472} a^{4} - \frac{1}{7736} a^{3} - \frac{45}{3868} a^{2} + \frac{23}{1934} a + \frac{11}{967}$, $\frac{1}{15843328} a^{36} - \frac{1}{15843328} a^{35} - \frac{1}{15843328} a^{34} + \frac{3}{15843328} a^{33} - \frac{1}{15843328} a^{32} - \frac{5}{15843328} a^{31} + \frac{7}{15843328} a^{30} + \frac{3}{15843328} a^{29} - \frac{17}{15843328} a^{28} + \frac{11}{15843328} a^{27} + \frac{23}{15843328} a^{26} - \frac{45}{15843328} a^{25} - \frac{1}{15843328} a^{24} + \frac{91}{15843328} a^{23} - \frac{6879}{16384} a^{22} - \frac{4795}{16384} a^{21} + \frac{2169}{16384} a^{20} + \frac{7421}{16384} a^{19} + \frac{4625}{16384} a^{18} - \frac{3083}{16384} a^{17} - \frac{6167}{16384} a^{16} - \frac{4051}{16384} a^{15} + \frac{1}{16384} a^{14} - \frac{1541}{7921664} a^{13} + \frac{287}{3960832} a^{12} + \frac{627}{1980416} a^{11} - \frac{457}{990208} a^{10} - \frac{85}{495104} a^{9} + \frac{271}{247552} a^{8} - \frac{93}{123776} a^{7} - \frac{89}{61888} a^{6} + \frac{91}{30944} a^{5} - \frac{1}{15472} a^{4} - \frac{45}{7736} a^{3} + \frac{23}{3868} a^{2} + \frac{11}{1934} a - \frac{17}{967}$, $\frac{1}{31686656} a^{37} - \frac{1}{31686656} a^{36} - \frac{1}{31686656} a^{35} + \frac{3}{31686656} a^{34} - \frac{1}{31686656} a^{33} - \frac{5}{31686656} a^{32} + \frac{7}{31686656} a^{31} + \frac{3}{31686656} a^{30} - \frac{17}{31686656} a^{29} + \frac{11}{31686656} a^{28} + \frac{23}{31686656} a^{27} - \frac{45}{31686656} a^{26} - \frac{1}{31686656} a^{25} + \frac{91}{31686656} a^{24} - \frac{89}{31686656} a^{23} + \frac{11589}{32768} a^{22} + \frac{2169}{32768} a^{21} + \frac{7421}{32768} a^{20} - \frac{11759}{32768} a^{19} - \frac{3083}{32768} a^{18} - \frac{6167}{32768} a^{17} + \frac{12333}{32768} a^{16} + \frac{1}{32768} a^{15} - \frac{1541}{15843328} a^{14} + \frac{287}{7921664} a^{13} + \frac{627}{3960832} a^{12} - \frac{457}{1980416} a^{11} - \frac{85}{990208} a^{10} + \frac{271}{495104} a^{9} - \frac{93}{247552} a^{8} - \frac{89}{123776} a^{7} + \frac{91}{61888} a^{6} - \frac{1}{30944} a^{5} - \frac{45}{15472} a^{4} + \frac{23}{7736} a^{3} + \frac{11}{3868} a^{2} - \frac{17}{1934} a + \frac{3}{967}$, $\frac{1}{63373312} a^{38} - \frac{1}{63373312} a^{37} - \frac{1}{63373312} a^{36} + \frac{3}{63373312} a^{35} - \frac{1}{63373312} a^{34} - \frac{5}{63373312} a^{33} + \frac{7}{63373312} a^{32} + \frac{3}{63373312} a^{31} - \frac{17}{63373312} a^{30} + \frac{11}{63373312} a^{29} + \frac{23}{63373312} a^{28} - \frac{45}{63373312} a^{27} - \frac{1}{63373312} a^{26} + \frac{91}{63373312} a^{25} - \frac{89}{63373312} a^{24} - \frac{93}{63373312} a^{23} + \frac{2169}{65536} a^{22} - \frac{25347}{65536} a^{21} + \frac{21009}{65536} a^{20} + \frac{29685}{65536} a^{19} - \frac{6167}{65536} a^{18} + \frac{12333}{65536} a^{17} + \frac{1}{65536} a^{16} - \frac{1541}{31686656} a^{15} + \frac{287}{15843328} a^{14} + \frac{627}{7921664} a^{13} - \frac{457}{3960832} a^{12} - \frac{85}{1980416} a^{11} + \frac{271}{990208} a^{10} - \frac{93}{495104} a^{9} - \frac{89}{247552} a^{8} + \frac{91}{123776} a^{7} - \frac{1}{61888} a^{6} - \frac{45}{30944} a^{5} + \frac{23}{15472} a^{4} + \frac{11}{7736} a^{3} - \frac{17}{3868} a^{2} + \frac{3}{1934} a + \frac{7}{967}$, $\frac{1}{126746624} a^{39} - \frac{1}{126746624} a^{38} - \frac{1}{126746624} a^{37} + \frac{3}{126746624} a^{36} - \frac{1}{126746624} a^{35} - \frac{5}{126746624} a^{34} + \frac{7}{126746624} a^{33} + \frac{3}{126746624} a^{32} - \frac{17}{126746624} a^{31} + \frac{11}{126746624} a^{30} + \frac{23}{126746624} a^{29} - \frac{45}{126746624} a^{28} - \frac{1}{126746624} a^{27} + \frac{91}{126746624} a^{26} - \frac{89}{126746624} a^{25} - \frac{93}{126746624} a^{24} + \frac{271}{126746624} a^{23} - \frac{25347}{131072} a^{22} + \frac{21009}{131072} a^{21} + \frac{29685}{131072} a^{20} + \frac{59369}{131072} a^{19} + \frac{12333}{131072} a^{18} + \frac{1}{131072} a^{17} - \frac{1541}{63373312} a^{16} + \frac{287}{31686656} a^{15} + \frac{627}{15843328} a^{14} - \frac{457}{7921664} a^{13} - \frac{85}{3960832} a^{12} + \frac{271}{1980416} a^{11} - \frac{93}{990208} a^{10} - \frac{89}{495104} a^{9} + \frac{91}{247552} a^{8} - \frac{1}{123776} a^{7} - \frac{45}{61888} a^{6} + \frac{23}{30944} a^{5} + \frac{11}{15472} a^{4} - \frac{17}{7736} a^{3} + \frac{3}{3868} a^{2} + \frac{7}{1934} a - \frac{5}{967}$, $\frac{1}{253493248} a^{40} - \frac{1}{253493248} a^{39} - \frac{1}{253493248} a^{38} + \frac{3}{253493248} a^{37} - \frac{1}{253493248} a^{36} - \frac{5}{253493248} a^{35} + \frac{7}{253493248} a^{34} + \frac{3}{253493248} a^{33} - \frac{17}{253493248} a^{32} + \frac{11}{253493248} a^{31} + \frac{23}{253493248} a^{30} - \frac{45}{253493248} a^{29} - \frac{1}{253493248} a^{28} + \frac{91}{253493248} a^{27} - \frac{89}{253493248} a^{26} - \frac{93}{253493248} a^{25} + \frac{271}{253493248} a^{24} - \frac{85}{253493248} a^{23} - \frac{110063}{262144} a^{22} - \frac{101387}{262144} a^{21} + \frac{59369}{262144} a^{20} - \frac{118739}{262144} a^{19} + \frac{1}{262144} a^{18} - \frac{1541}{126746624} a^{17} + \frac{287}{63373312} a^{16} + \frac{627}{31686656} a^{15} - \frac{457}{15843328} a^{14} - \frac{85}{7921664} a^{13} + \frac{271}{3960832} a^{12} - \frac{93}{1980416} a^{11} - \frac{89}{990208} a^{10} + \frac{91}{495104} a^{9} - \frac{1}{247552} a^{8} - \frac{45}{123776} a^{7} + \frac{23}{61888} a^{6} + \frac{11}{30944} a^{5} - \frac{17}{15472} a^{4} + \frac{3}{7736} a^{3} + \frac{7}{3868} a^{2} - \frac{5}{1934} a - \frac{1}{967}$, $\frac{1}{506986496} a^{41} - \frac{1}{506986496} a^{40} - \frac{1}{506986496} a^{39} + \frac{3}{506986496} a^{38} - \frac{1}{506986496} a^{37} - \frac{5}{506986496} a^{36} + \frac{7}{506986496} a^{35} + \frac{3}{506986496} a^{34} - \frac{17}{506986496} a^{33} + \frac{11}{506986496} a^{32} + \frac{23}{506986496} a^{31} - \frac{45}{506986496} a^{30} - \frac{1}{506986496} a^{29} + \frac{91}{506986496} a^{28} - \frac{89}{506986496} a^{27} - \frac{93}{506986496} a^{26} + \frac{271}{506986496} a^{25} - \frac{85}{506986496} a^{24} - \frac{457}{506986496} a^{23} - \frac{101387}{524288} a^{22} - \frac{202775}{524288} a^{21} - \frac{118739}{524288} a^{20} + \frac{1}{524288} a^{19} - \frac{1541}{253493248} a^{18} + \frac{287}{126746624} a^{17} + \frac{627}{63373312} a^{16} - \frac{457}{31686656} a^{15} - \frac{85}{15843328} a^{14} + \frac{271}{7921664} a^{13} - \frac{93}{3960832} a^{12} - \frac{89}{1980416} a^{11} + \frac{91}{990208} a^{10} - \frac{1}{495104} a^{9} - \frac{45}{247552} a^{8} + \frac{23}{123776} a^{7} + \frac{11}{61888} a^{6} - \frac{17}{30944} a^{5} + \frac{3}{15472} a^{4} + \frac{7}{7736} a^{3} - \frac{5}{3868} a^{2} - \frac{1}{1934} a + \frac{3}{967}$, $\frac{1}{1013972992} a^{42} - \frac{1}{1013972992} a^{41} - \frac{1}{1013972992} a^{40} + \frac{3}{1013972992} a^{39} - \frac{1}{1013972992} a^{38} - \frac{5}{1013972992} a^{37} + \frac{7}{1013972992} a^{36} + \frac{3}{1013972992} a^{35} - \frac{17}{1013972992} a^{34} + \frac{11}{1013972992} a^{33} + \frac{23}{1013972992} a^{32} - \frac{45}{1013972992} a^{31} - \frac{1}{1013972992} a^{30} + \frac{91}{1013972992} a^{29} - \frac{89}{1013972992} a^{28} - \frac{93}{1013972992} a^{27} + \frac{271}{1013972992} a^{26} - \frac{85}{1013972992} a^{25} - \frac{457}{1013972992} a^{24} + \frac{627}{1013972992} a^{23} - \frac{202775}{1048576} a^{22} + \frac{405549}{1048576} a^{21} + \frac{1}{1048576} a^{20} - \frac{1541}{506986496} a^{19} + \frac{287}{253493248} a^{18} + \frac{627}{126746624} a^{17} - \frac{457}{63373312} a^{16} - \frac{85}{31686656} a^{15} + \frac{271}{15843328} a^{14} - \frac{93}{7921664} a^{13} - \frac{89}{3960832} a^{12} + \frac{91}{1980416} a^{11} - \frac{1}{990208} a^{10} - \frac{45}{495104} a^{9} + \frac{23}{247552} a^{8} + \frac{11}{123776} a^{7} - \frac{17}{61888} a^{6} + \frac{3}{30944} a^{5} + \frac{7}{15472} a^{4} - \frac{5}{7736} a^{3} - \frac{1}{3868} a^{2} + \frac{3}{1934} a - \frac{1}{967}$, $\frac{1}{2027945984} a^{43} - \frac{1}{2027945984} a^{42} - \frac{1}{2027945984} a^{41} + \frac{3}{2027945984} a^{40} - \frac{1}{2027945984} a^{39} - \frac{5}{2027945984} a^{38} + \frac{7}{2027945984} a^{37} + \frac{3}{2027945984} a^{36} - \frac{17}{2027945984} a^{35} + \frac{11}{2027945984} a^{34} + \frac{23}{2027945984} a^{33} - \frac{45}{2027945984} a^{32} - \frac{1}{2027945984} a^{31} + \frac{91}{2027945984} a^{30} - \frac{89}{2027945984} a^{29} - \frac{93}{2027945984} a^{28} + \frac{271}{2027945984} a^{27} - \frac{85}{2027945984} a^{26} - \frac{457}{2027945984} a^{25} + \frac{627}{2027945984} a^{24} + \frac{287}{2027945984} a^{23} + \frac{405549}{2097152} a^{22} + \frac{1}{2097152} a^{21} - \frac{1541}{1013972992} a^{20} + \frac{287}{506986496} a^{19} + \frac{627}{253493248} a^{18} - \frac{457}{126746624} a^{17} - \frac{85}{63373312} a^{16} + \frac{271}{31686656} a^{15} - \frac{93}{15843328} a^{14} - \frac{89}{7921664} a^{13} + \frac{91}{3960832} a^{12} - \frac{1}{1980416} a^{11} - \frac{45}{990208} a^{10} + \frac{23}{495104} a^{9} + \frac{11}{247552} a^{8} - \frac{17}{123776} a^{7} + \frac{3}{61888} a^{6} + \frac{7}{30944} a^{5} - \frac{5}{15472} a^{4} - \frac{1}{7736} a^{3} + \frac{3}{3868} a^{2} - \frac{1}{1934} a - \frac{1}{967}$

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:  \( \frac{287}{2027945984} a^{43} + \frac{287}{2027945984} a^{42} - \frac{861}{2027945984} a^{41} + \frac{287}{2027945984} a^{40} + \frac{1435}{2027945984} a^{39} - \frac{2009}{2027945984} a^{38} - \frac{861}{2027945984} a^{37} + \frac{4879}{2027945984} a^{36} - \frac{3157}{2027945984} a^{35} - \frac{6601}{2027945984} a^{34} + \frac{12915}{2027945984} a^{33} + \frac{287}{2027945984} a^{32} - \frac{26117}{2027945984} a^{31} + \frac{25543}{2027945984} a^{30} + \frac{26691}{2027945984} a^{29} - \frac{77777}{2027945984} a^{28} + \frac{24395}{2027945984} a^{27} + \frac{131159}{2027945984} a^{26} - \frac{179949}{2027945984} a^{25} - \frac{82369}{2027945984} a^{24} + \frac{442267}{2027945984} a^{23} - \frac{287}{2097152} a^{22} + \frac{1541}{2097152} a^{21} - \frac{82369}{506986496} a^{20} - \frac{179949}{253493248} a^{19} + \frac{131159}{126746624} a^{18} + \frac{24395}{63373312} a^{17} - \frac{77777}{31686656} a^{16} + \frac{26691}{15843328} a^{15} + \frac{25543}{7921664} a^{14} - \frac{26117}{3960832} a^{13} + \frac{287}{1980416} a^{12} + \frac{12915}{990208} a^{11} - \frac{6601}{495104} a^{10} - \frac{3157}{247552} a^{9} + \frac{4879}{123776} a^{8} - \frac{861}{61888} a^{7} - \frac{2009}{30944} a^{6} + \frac{1435}{15472} a^{5} + \frac{287}{7736} a^{4} - \frac{861}{3868} a^{3} + \frac{287}{1934} a^{2} + \frac{287}{967} a - \frac{574}{967} \) (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

\(\Q(\sqrt{161}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-7}, \sqrt{-23})\), \(\Q(\zeta_{23})^+\), 22.22.78048218870425324004237696277333187889.1, 22.0.3393400820453274956705986794666660343.1, \(\Q(\zeta_{23})\)

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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ R $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ 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
7Data not computed
23Data not computed