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Properties

Label 44.44.191...489.1
Degree $44$
Signature $[44, 0]$
Discriminant $1.912\times 10^{86}$
Root discriminant $91.40$
Ramified primes $3, 7, 23$
Class number not computed
Class group not computed
Galois group $C_2\times C_{22}$ (as 44T2)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 - 86*x^42 + 81*x^41 + 3445*x^40 - 3040*x^39 - 85351*x^38 + 70151*x^37 + 1464534*x^36 - 1113779*x^35 - 18468767*x^34 + 12899872*x^33 + 177261061*x^32 - 112761701*x^31 - 1323060098*x^30 + 759251593*x^29 + 7780382701*x^28 - 3984124736*x^27 - 36301909279*x^26 + 16381285599*x^25 + 134698547590*x^24 - 52792119595*x^23 - 396744216375*x^22 + 132783613203*x^21 + 922279857882*x^20 - 258361552805*x^19 - 1675313579545*x^18 + 383501034280*x^17 + 2343182319195*x^16 - 425622641659*x^15 - 2471777086526*x^14 + 343273729047*x^13 + 1911152162483*x^12 - 192962821056*x^11 - 1040820630209*x^10 + 70437336577*x^9 + 377125555258*x^8 - 13999395253*x^7 - 83413545257*x^6 + 289196448*x^5 + 9759539347*x^4 + 438024589*x^3 - 453637806*x^2 - 49033617*x - 368597)
 
gp: K = bnfinit(x^44 - x^43 - 86*x^42 + 81*x^41 + 3445*x^40 - 3040*x^39 - 85351*x^38 + 70151*x^37 + 1464534*x^36 - 1113779*x^35 - 18468767*x^34 + 12899872*x^33 + 177261061*x^32 - 112761701*x^31 - 1323060098*x^30 + 759251593*x^29 + 7780382701*x^28 - 3984124736*x^27 - 36301909279*x^26 + 16381285599*x^25 + 134698547590*x^24 - 52792119595*x^23 - 396744216375*x^22 + 132783613203*x^21 + 922279857882*x^20 - 258361552805*x^19 - 1675313579545*x^18 + 383501034280*x^17 + 2343182319195*x^16 - 425622641659*x^15 - 2471777086526*x^14 + 343273729047*x^13 + 1911152162483*x^12 - 192962821056*x^11 - 1040820630209*x^10 + 70437336577*x^9 + 377125555258*x^8 - 13999395253*x^7 - 83413545257*x^6 + 289196448*x^5 + 9759539347*x^4 + 438024589*x^3 - 453637806*x^2 - 49033617*x - 368597, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-368597, -49033617, -453637806, 438024589, 9759539347, 289196448, -83413545257, -13999395253, 377125555258, 70437336577, -1040820630209, -192962821056, 1911152162483, 343273729047, -2471777086526, -425622641659, 2343182319195, 383501034280, -1675313579545, -258361552805, 922279857882, 132783613203, -396744216375, -52792119595, 134698547590, 16381285599, -36301909279, -3984124736, 7780382701, 759251593, -1323060098, -112761701, 177261061, 12899872, -18468767, -1113779, 1464534, 70151, -85351, -3040, 3445, 81, -86, -1, 1]);
 

\( x^{44} - x^{43} - 86 x^{42} + 81 x^{41} + 3445 x^{40} - 3040 x^{39} - 85351 x^{38} + 70151 x^{37} + 1464534 x^{36} - 1113779 x^{35} - 18468767 x^{34} + 12899872 x^{33} + 177261061 x^{32} - 112761701 x^{31} - 1323060098 x^{30} + 759251593 x^{29} + 7780382701 x^{28} - 3984124736 x^{27} - 36301909279 x^{26} + 16381285599 x^{25} + 134698547590 x^{24} - 52792119595 x^{23} - 396744216375 x^{22} + 132783613203 x^{21} + 922279857882 x^{20} - 258361552805 x^{19} - 1675313579545 x^{18} + 383501034280 x^{17} + 2343182319195 x^{16} - 425622641659 x^{15} - 2471777086526 x^{14} + 343273729047 x^{13} + 1911152162483 x^{12} - 192962821056 x^{11} - 1040820630209 x^{10} + 70437336577 x^{9} + 377125555258 x^{8} - 13999395253 x^{7} - 83413545257 x^{6} + 289196448 x^{5} + 9759539347 x^{4} + 438024589 x^{3} - 453637806 x^{2} - 49033617 x - 368597 \)

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

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[44, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(191\!\cdots\!489\)\(\medspace = 3^{22}\cdot 7^{22}\cdot 23^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $91.40$
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:  $3, 7, 23$
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:  \(483=3\cdot 7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{483}(1,·)$, $\chi_{483}(260,·)$, $\chi_{483}(134,·)$, $\chi_{483}(400,·)$, $\chi_{483}(146,·)$, $\chi_{483}(281,·)$, $\chi_{483}(155,·)$, $\chi_{483}(412,·)$, $\chi_{483}(286,·)$, $\chi_{483}(160,·)$, $\chi_{483}(34,·)$, $\chi_{483}(167,·)$, $\chi_{483}(41,·)$, $\chi_{483}(428,·)$, $\chi_{483}(176,·)$, $\chi_{483}(433,·)$, $\chi_{483}(181,·)$, $\chi_{483}(440,·)$, $\chi_{483}(188,·)$, $\chi_{483}(190,·)$, $\chi_{483}(64,·)$, $\chi_{483}(454,·)$, $\chi_{483}(113,·)$, $\chi_{483}(76,·)$, $\chi_{483}(461,·)$, $\chi_{483}(463,·)$, $\chi_{483}(209,·)$, $\chi_{483}(335,·)$, $\chi_{483}(211,·)$, $\chi_{483}(85,·)$, $\chi_{483}(470,·)$, $\chi_{483}(344,·)$, $\chi_{483}(218,·)$, $\chi_{483}(475,·)$, $\chi_{483}(97,·)$, $\chi_{483}(358,·)$, $\chi_{483}(104,·)$, $\chi_{483}(365,·)$, $\chi_{483}(232,·)$, $\chi_{483}(244,·)$, $\chi_{483}(62,·)$, $\chi_{483}(169,·)$, $\chi_{483}(377,·)$, $\chi_{483}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{967} a^{23} - \frac{46}{967} a^{21} - \frac{47}{967} a^{19} + \frac{149}{967} a^{17} - \frac{354}{967} a^{15} - \frac{282}{967} a^{13} + \frac{180}{967} a^{11} + \frac{199}{967} a^{9} + \frac{148}{967} a^{7} - \frac{421}{967} a^{5} - \frac{168}{967} a^{3} + \frac{279}{967} a - \frac{181}{967}$, $\frac{1}{967} a^{24} - \frac{46}{967} a^{22} - \frac{47}{967} a^{20} + \frac{149}{967} a^{18} - \frac{354}{967} a^{16} - \frac{282}{967} a^{14} + \frac{180}{967} a^{12} + \frac{199}{967} a^{10} + \frac{148}{967} a^{8} - \frac{421}{967} a^{6} - \frac{168}{967} a^{4} + \frac{279}{967} a^{2} - \frac{181}{967} a$, $\frac{1}{967} a^{25} - \frac{229}{967} a^{21} - \frac{79}{967} a^{19} - \frac{269}{967} a^{17} - \frac{127}{967} a^{15} - \frac{221}{967} a^{13} - \frac{224}{967} a^{11} - \frac{368}{967} a^{9} - \frac{382}{967} a^{7} - \frac{194}{967} a^{5} + \frac{287}{967} a^{3} - \frac{181}{967} a^{2} + \frac{263}{967} a + \frac{377}{967}$, $\frac{1}{967} a^{26} - \frac{229}{967} a^{22} - \frac{79}{967} a^{20} - \frac{269}{967} a^{18} - \frac{127}{967} a^{16} - \frac{221}{967} a^{14} - \frac{224}{967} a^{12} - \frac{368}{967} a^{10} - \frac{382}{967} a^{8} - \frac{194}{967} a^{6} + \frac{287}{967} a^{4} - \frac{181}{967} a^{3} + \frac{263}{967} a^{2} + \frac{377}{967} a$, $\frac{1}{967} a^{27} + \frac{24}{967} a^{21} - \frac{395}{967} a^{19} + \frac{149}{967} a^{17} - \frac{59}{967} a^{15} - \frac{13}{967} a^{13} + \frac{238}{967} a^{11} - \frac{260}{967} a^{9} - \frac{147}{967} a^{7} - \frac{389}{967} a^{5} - \frac{181}{967} a^{4} + \frac{471}{967} a^{3} + \frac{377}{967} a^{2} + \frac{69}{967} a + \frac{132}{967}$, $\frac{1}{967} a^{28} + \frac{24}{967} a^{22} - \frac{395}{967} a^{20} + \frac{149}{967} a^{18} - \frac{59}{967} a^{16} - \frac{13}{967} a^{14} + \frac{238}{967} a^{12} - \frac{260}{967} a^{10} - \frac{147}{967} a^{8} - \frac{389}{967} a^{6} - \frac{181}{967} a^{5} + \frac{471}{967} a^{4} + \frac{377}{967} a^{3} + \frac{69}{967} a^{2} + \frac{132}{967} a$, $\frac{1}{967} a^{29} - \frac{258}{967} a^{21} + \frac{310}{967} a^{19} + \frac{233}{967} a^{17} - \frac{220}{967} a^{15} + \frac{237}{967} a^{13} + \frac{255}{967} a^{11} - \frac{88}{967} a^{9} - \frac{73}{967} a^{7} - \frac{181}{967} a^{6} - \frac{62}{967} a^{5} + \frac{377}{967} a^{4} + \frac{233}{967} a^{3} + \frac{132}{967} a^{2} + \frac{73}{967} a + \frac{476}{967}$, $\frac{1}{967} a^{30} - \frac{258}{967} a^{22} + \frac{310}{967} a^{20} + \frac{233}{967} a^{18} - \frac{220}{967} a^{16} + \frac{237}{967} a^{14} + \frac{255}{967} a^{12} - \frac{88}{967} a^{10} - \frac{73}{967} a^{8} - \frac{181}{967} a^{7} - \frac{62}{967} a^{6} + \frac{377}{967} a^{5} + \frac{233}{967} a^{4} + \frac{132}{967} a^{3} + \frac{73}{967} a^{2} + \frac{476}{967} a$, $\frac{1}{967} a^{31} + \frac{46}{967} a^{21} - \frac{289}{967} a^{19} - \frac{458}{967} a^{17} - \frac{197}{967} a^{15} + \frac{24}{967} a^{13} - \frac{64}{967} a^{11} + \frac{18}{967} a^{9} - \frac{181}{967} a^{8} + \frac{409}{967} a^{7} + \frac{377}{967} a^{6} - \frac{81}{967} a^{5} + \frac{132}{967} a^{4} + \frac{244}{967} a^{3} + \frac{476}{967} a^{2} + \frac{424}{967} a - \frac{282}{967}$, $\frac{1}{967} a^{32} + \frac{46}{967} a^{22} - \frac{289}{967} a^{20} - \frac{458}{967} a^{18} - \frac{197}{967} a^{16} + \frac{24}{967} a^{14} - \frac{64}{967} a^{12} + \frac{18}{967} a^{10} - \frac{181}{967} a^{9} + \frac{409}{967} a^{8} + \frac{377}{967} a^{7} - \frac{81}{967} a^{6} + \frac{132}{967} a^{5} + \frac{244}{967} a^{4} + \frac{476}{967} a^{3} + \frac{424}{967} a^{2} - \frac{282}{967} a$, $\frac{1}{967} a^{33} - \frac{107}{967} a^{21} - \frac{230}{967} a^{19} - \frac{282}{967} a^{17} - \frac{131}{967} a^{15} + \frac{337}{967} a^{13} + \frac{441}{967} a^{11} - \frac{181}{967} a^{10} - \frac{42}{967} a^{9} + \frac{377}{967} a^{8} - \frac{120}{967} a^{7} + \frac{132}{967} a^{6} + \frac{270}{967} a^{5} + \frac{476}{967} a^{4} + \frac{416}{967} a^{3} - \frac{282}{967} a^{2} - \frac{263}{967} a - \frac{377}{967}$, $\frac{1}{18005734367} a^{34} - \frac{591175}{18005734367} a^{33} + \frac{8219207}{18005734367} a^{32} - \frac{4614865}{18005734367} a^{31} - \frac{7492431}{18005734367} a^{30} + \frac{8807653}{18005734367} a^{29} + \frac{1722207}{18005734367} a^{28} + \frac{6507841}{18005734367} a^{27} - \frac{2886529}{18005734367} a^{26} + \frac{5395227}{18005734367} a^{25} - \frac{7252309}{18005734367} a^{24} + \frac{6524189}{18005734367} a^{23} - \frac{8116723272}{18005734367} a^{22} - \frac{7617802541}{18005734367} a^{21} + \frac{2967351794}{18005734367} a^{20} + \frac{6954293200}{18005734367} a^{19} + \frac{8986155566}{18005734367} a^{18} - \frac{1318943358}{18005734367} a^{17} - \frac{5376394790}{18005734367} a^{16} - \frac{6956511157}{18005734367} a^{15} + \frac{526187471}{18005734367} a^{14} - \frac{7729633096}{18005734367} a^{13} - \frac{8236571390}{18005734367} a^{12} - \frac{6473597624}{18005734367} a^{11} - \frac{8830373379}{18005734367} a^{10} - \frac{1822289900}{18005734367} a^{9} + \frac{4660704427}{18005734367} a^{8} - \frac{3095868076}{18005734367} a^{7} + \frac{6437991265}{18005734367} a^{6} - \frac{3128984668}{18005734367} a^{5} - \frac{5591973051}{18005734367} a^{4} + \frac{5300710589}{18005734367} a^{3} - \frac{8347844180}{18005734367} a^{2} - \frac{7999597184}{18005734367} a + \frac{5880106311}{18005734367}$, $\frac{1}{18005734367} a^{35} + \frac{2891151}{18005734367} a^{33} + \frac{6392008}{18005734367} a^{32} - \frac{5698688}{18005734367} a^{31} + \frac{7084706}{18005734367} a^{30} + \frac{6077847}{18005734367} a^{29} - \frac{1739413}{18005734367} a^{28} + \frac{1326428}{18005734367} a^{27} - \frac{8685904}{18005734367} a^{26} + \frac{5979523}{18005734367} a^{25} - \frac{1487832}{18005734367} a^{24} + \frac{6541902}{18005734367} a^{23} + \frac{4188803716}{18005734367} a^{22} + \frac{4295754719}{18005734367} a^{21} - \frac{3643735828}{18005734367} a^{20} - \frac{6229789542}{18005734367} a^{19} + \frac{5986636351}{18005734367} a^{18} + \frac{4562480385}{18005734367} a^{17} + \frac{984673024}{18005734367} a^{16} + \frac{7974093305}{18005734367} a^{15} + \frac{3838596361}{18005734367} a^{14} + \frac{2090331415}{18005734367} a^{13} + \frac{411576391}{18005734367} a^{12} + \frac{8836545999}{18005734367} a^{11} - \frac{6234614076}{18005734367} a^{10} + \frac{7693860991}{18005734367} a^{9} + \frac{1431546227}{18005734367} a^{8} - \frac{3071692058}{18005734367} a^{7} + \frac{6807196682}{18005734367} a^{6} - \frac{8559645400}{18005734367} a^{5} + \frac{8677627270}{18005734367} a^{4} + \frac{2949437404}{18005734367} a^{3} + \frac{5566756153}{18005734367} a^{2} + \frac{244208981}{18005734367} a + \frac{1380731697}{18005734367}$, $\frac{1}{18005734367} a^{36} - \frac{2905759}{18005734367} a^{33} - \frac{4061152}{18005734367} a^{32} - \frac{7762028}{18005734367} a^{31} + \frac{483181}{18005734367} a^{30} + \frac{4181745}{18005734367} a^{29} - \frac{5695223}{18005734367} a^{28} - \frac{5196425}{18005734367} a^{27} + \frac{9298212}{18005734367} a^{26} + \frac{4256606}{18005734367} a^{25} + \frac{7560898}{18005734367} a^{24} - \frac{3718440}{18005734367} a^{23} + \frac{320156472}{18005734367} a^{22} - \frac{5478291523}{18005734367} a^{21} + \frac{7168819784}{18005734367} a^{20} - \frac{1708076999}{18005734367} a^{19} + \frac{8798946849}{18005734367} a^{18} + \frac{429821056}{18005734367} a^{17} + \frac{3131789313}{18005734367} a^{16} + \frac{8764057688}{18005734367} a^{15} + \frac{8223972137}{18005734367} a^{14} - \frac{5117701182}{18005734367} a^{13} + \frac{1608094027}{18005734367} a^{12} - \frac{432570598}{18005734367} a^{11} + \frac{6073271067}{18005734367} a^{10} + \frac{3909560052}{18005734367} a^{9} + \frac{8961572383}{18005734367} a^{8} - \frac{7474250380}{18005734367} a^{7} + \frac{2076432229}{18005734367} a^{6} - \frac{5795486410}{18005734367} a^{5} + \frac{5936703895}{18005734367} a^{4} - \frac{1192062182}{18005734367} a^{3} - \frac{6637331363}{18005734367} a^{2} - \frac{4825927838}{18005734367} a - \frac{8732205726}{18005734367}$, $\frac{1}{18005734367} a^{37} + \frac{9125479}{18005734367} a^{33} - \frac{6279756}{18005734367} a^{32} + \frac{6509615}{18005734367} a^{31} - \frac{8734360}{18005734367} a^{30} + \frac{3609934}{18005734367} a^{29} + \frac{3933531}{18005734367} a^{28} - \frac{2396245}{18005734367} a^{27} + \frac{9277550}{18005734367} a^{26} - \frac{4018357}{18005734367} a^{25} + \frac{2476382}{18005734367} a^{24} - \frac{4866021}{18005734367} a^{23} - \frac{8756667875}{18005734367} a^{22} + \frac{2613994159}{18005734367} a^{21} - \frac{1224129587}{18005734367} a^{20} - \frac{2313735423}{18005734367} a^{19} - \frac{4266003562}{18005734367} a^{18} + \frac{5286048246}{18005734367} a^{17} + \frac{7251021131}{18005734367} a^{16} - \frac{1225971537}{18005734367} a^{15} - \frac{3694278137}{18005734367} a^{14} + \frac{1068500791}{18005734367} a^{13} - \frac{7734120736}{18005734367} a^{12} + \frac{837459922}{18005734367} a^{11} - \frac{3880875919}{18005734367} a^{10} + \frac{6321693380}{18005734367} a^{9} + \frac{599308239}{18005734367} a^{8} + \frac{5824256901}{18005734367} a^{7} + \frac{3595148396}{18005734367} a^{6} + \frac{1482750779}{18005734367} a^{5} + \frac{7873587205}{18005734367} a^{4} + \frac{7201261728}{18005734367} a^{3} + \frac{7519856446}{18005734367} a^{2} - \frac{3015523811}{18005734367} a - \frac{5936071860}{18005734367}$, $\frac{1}{18005734367} a^{38} - \frac{7586857}{18005734367} a^{33} + \frac{4864371}{18005734367} a^{32} + \frac{1814300}{18005734367} a^{31} + \frac{6562056}{18005734367} a^{30} - \frac{6801359}{18005734367} a^{29} + \frac{4801230}{18005734367} a^{28} + \frac{244905}{18005734367} a^{27} + \frac{130590}{18005734367} a^{26} - \frac{964432}{18005734367} a^{25} - \frac{4928657}{18005734367} a^{24} + \frac{6133871}{18005734367} a^{23} + \frac{2551964881}{18005734367} a^{22} + \frac{756213003}{18005734367} a^{21} - \frac{8965561854}{18005734367} a^{20} + \frac{7397365223}{18005734367} a^{19} + \frac{1908001736}{18005734367} a^{18} - \frac{7007048352}{18005734367} a^{17} + \frac{6322353875}{18005734367} a^{16} + \frac{8442571972}{18005734367} a^{15} - \frac{1991824458}{18005734367} a^{14} + \frac{7844400165}{18005734367} a^{13} - \frac{7702268240}{18005734367} a^{12} - \frac{2312474263}{18005734367} a^{11} + \frac{8868333767}{18005734367} a^{10} - \frac{6687850982}{18005734367} a^{9} - \frac{7855350190}{18005734367} a^{8} - \frac{4503376574}{18005734367} a^{7} - \frac{8665280315}{18005734367} a^{6} - \frac{6424255271}{18005734367} a^{5} - \frac{3377377461}{18005734367} a^{4} - \frac{7277758159}{18005734367} a^{3} - \frac{6432875800}{18005734367} a^{2} - \frac{5736567751}{18005734367} a + \frac{3250209210}{18005734367}$, $\frac{1}{18005734367} a^{39} + \frac{4213472}{18005734367} a^{33} - \frac{4580442}{18005734367} a^{32} - \frac{5518909}{18005734367} a^{31} + \frac{914888}{18005734367} a^{30} + \frac{188146}{18005734367} a^{29} + \frac{8272986}{18005734367} a^{28} + \frac{8016888}{18005734367} a^{27} + \frac{187340}{18005734367} a^{26} + \frac{4324381}{18005734367} a^{25} + \frac{3281033}{18005734367} a^{24} - \frac{8957184}{18005734367} a^{23} - \frac{6744250151}{18005734367} a^{22} + \frac{7965373416}{18005734367} a^{21} + \frac{546580484}{18005734367} a^{20} - \frac{2922409241}{18005734367} a^{19} - \frac{3513942863}{18005734367} a^{18} - \frac{5095278699}{18005734367} a^{17} + \frac{4174270369}{18005734367} a^{16} - \frac{368972889}{18005734367} a^{15} + \frac{7145370504}{18005734367} a^{14} + \frac{7020692573}{18005734367} a^{13} - \frac{8468200194}{18005734367} a^{12} + \frac{4384435815}{18005734367} a^{11} - \frac{2277353109}{18005734367} a^{10} - \frac{4807282016}{18005734367} a^{9} + \frac{3921419956}{18005734367} a^{8} + \frac{7824816414}{18005734367} a^{7} + \frac{5111926715}{18005734367} a^{6} - \frac{8825424399}{18005734367} a^{5} - \frac{3799691411}{18005734367} a^{4} + \frac{3649891214}{18005734367} a^{3} - \frac{5608901741}{18005734367} a^{2} - \frac{3068694642}{18005734367} a + \frac{7934826520}{18005734367}$, $\frac{1}{18005734367} a^{40} - \frac{4039416}{18005734367} a^{33} - \frac{8779130}{18005734367} a^{32} - \frac{5023107}{18005734367} a^{31} - \frac{5861847}{18005734367} a^{30} - \frac{4247391}{18005734367} a^{29} - \frac{3044307}{18005734367} a^{28} + \frac{5091415}{18005734367} a^{27} + \frac{7794291}{18005734367} a^{26} + \frac{5355548}{18005734367} a^{25} + \frac{3530976}{18005734367} a^{24} - \frac{4550071}{18005734367} a^{23} + \frac{7997192356}{18005734367} a^{22} + \frac{3172132204}{18005734367} a^{21} + \frac{1518571401}{18005734367} a^{20} - \frac{5809972218}{18005734367} a^{19} - \frac{2000573561}{18005734367} a^{18} - \frac{7895674988}{18005734367} a^{17} + \frac{8571208629}{18005734367} a^{16} + \frac{5277511649}{18005734367} a^{15} + \frac{7715304425}{18005734367} a^{14} + \frac{3170224994}{18005734367} a^{13} - \frac{1147425478}{18005734367} a^{12} - \frac{5710452540}{18005734367} a^{11} - \frac{7969985851}{18005734367} a^{10} + \frac{645830128}{18005734367} a^{9} - \frac{7711633309}{18005734367} a^{8} + \frac{1259481103}{18005734367} a^{7} - \frac{391016636}{18005734367} a^{6} + \frac{5394939914}{18005734367} a^{5} + \frac{4744793178}{18005734367} a^{4} + \frac{8124331432}{18005734367} a^{3} + \frac{5293649129}{18005734367} a^{2} - \frac{909048877}{18005734367} a - \frac{2955114152}{18005734367}$, $\frac{1}{18005734367} a^{41} - \frac{6995082}{18005734367} a^{33} + \frac{4606553}{18005734367} a^{32} + \frac{6787850}{18005734367} a^{31} + \frac{1355301}{18005734367} a^{30} + \frac{4394033}{18005734367} a^{29} + \frac{3686716}{18005734367} a^{28} - \frac{3195447}{18005734367} a^{27} - \frac{9306321}{18005734367} a^{26} + \frac{2422782}{18005734367} a^{25} + \frac{8811082}{18005734367} a^{24} + \frac{7276411}{18005734367} a^{23} + \frac{7778147674}{18005734367} a^{22} - \frac{5926347193}{18005734367} a^{21} + \frac{4292606763}{18005734367} a^{20} + \frac{2068227716}{18005734367} a^{19} - \frac{57725774}{18005734367} a^{18} + \frac{6845099410}{18005734367} a^{17} + \frac{1407754133}{18005734367} a^{16} + \frac{7638905808}{18005734367} a^{15} + \frac{1356015547}{18005734367} a^{14} - \frac{1644701161}{18005734367} a^{13} + \frac{4885873290}{18005734367} a^{12} + \frac{5371344226}{18005734367} a^{11} + \frac{6325121466}{18005734367} a^{10} + \frac{8935538223}{18005734367} a^{9} + \frac{7587603073}{18005734367} a^{8} + \frac{4117245630}{18005734367} a^{7} + \frac{3465280453}{18005734367} a^{6} + \frac{8064681332}{18005734367} a^{5} + \frac{3316668054}{18005734367} a^{4} + \frac{6748597825}{18005734367} a^{3} - \frac{4982589257}{18005734367} a^{2} - \frac{4000887394}{18005734367} a - \frac{3544487360}{18005734367}$, $\frac{1}{18005734367} a^{42} - \frac{8415310}{18005734367} a^{33} - \frac{7784700}{18005734367} a^{32} + \frac{609845}{18005734367} a^{31} + \frac{3203185}{18005734367} a^{30} - \frac{1736724}{18005734367} a^{29} + \frac{3866743}{18005734367} a^{28} - \frac{95370}{18005734367} a^{27} - \frac{725809}{18005734367} a^{26} + \frac{7611062}{18005734367} a^{25} + \frac{6693960}{18005734367} a^{24} - \frac{5559399}{18005734367} a^{23} - \frac{8171795}{18005734367} a^{22} - \frac{960080681}{18005734367} a^{21} + \frac{4508740552}{18005734367} a^{20} + \frac{1545375374}{18005734367} a^{19} + \frac{8479026244}{18005734367} a^{18} - \frac{8061593078}{18005734367} a^{17} - \frac{6750371537}{18005734367} a^{16} + \frac{2869758516}{18005734367} a^{15} - \frac{1033755832}{18005734367} a^{14} - \frac{4934748138}{18005734367} a^{13} + \frac{7889459308}{18005734367} a^{12} - \frac{1577976098}{18005734367} a^{11} - \frac{4200062249}{18005734367} a^{10} + \frac{8138237792}{18005734367} a^{9} + \frac{6372420551}{18005734367} a^{8} + \frac{3813257769}{18005734367} a^{7} + \frac{6204503434}{18005734367} a^{6} - \frac{5075895478}{18005734367} a^{5} + \frac{3377582715}{18005734367} a^{4} + \frac{5270018394}{18005734367} a^{3} + \frac{457218847}{18005734367} a^{2} + \frac{8686783840}{18005734367} a + \frac{3126775714}{18005734367}$, $\frac{1}{18005734367} a^{43} - \frac{1990971}{18005734367} a^{33} - \frac{391816}{18005734367} a^{32} + \frac{3758901}{18005734367} a^{31} - \frac{2475566}{18005734367} a^{30} + \frac{9018397}{18005734367} a^{29} + \frac{2586857}{18005734367} a^{28} + \frac{5601314}{18005734367} a^{27} - \frac{8292976}{18005734367} a^{26} + \frac{3171583}{18005734367} a^{25} - \frac{6683343}{18005734367} a^{24} + \frac{5598220}{18005734367} a^{23} + \frac{3903195380}{18005734367} a^{22} - \frac{643848132}{18005734367} a^{21} + \frac{2418825435}{18005734367} a^{20} - \frac{2777454018}{18005734367} a^{19} + \frac{4620699613}{18005734367} a^{18} + \frac{2209981651}{18005734367} a^{17} - \frac{4324821486}{18005734367} a^{16} + \frac{5382131895}{18005734367} a^{15} - \frac{8981316434}{18005734367} a^{14} - \frac{7238956882}{18005734367} a^{13} + \frac{2416364515}{18005734367} a^{12} + \frac{3169087318}{18005734367} a^{11} + \frac{6027935631}{18005734367} a^{10} + \frac{4960877003}{18005734367} a^{9} - \frac{4844876804}{18005734367} a^{8} - \frac{1326830891}{18005734367} a^{7} + \frac{3174503826}{18005734367} a^{6} - \frac{8752341093}{18005734367} a^{5} - \frac{8365286509}{18005734367} a^{4} + \frac{8741680307}{18005734367} a^{3} + \frac{2057324183}{18005734367} a^{2} + \frac{8312953902}{18005734367} a + \frac{445165358}{18005734367}$

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:  $43$
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

$C_2\times C_{22}$ (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

\(\Q(\sqrt{161}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{69}) \), \(\Q(\sqrt{21}, \sqrt{69})\), \(\Q(\zeta_{23})^+\), 22.22.78048218870425324004237696277333187889.1, 22.22.601130775140836298755595442714814879781421.1, \(\Q(\zeta_{69})^+\)

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 $22^{2}$ R ${\href{/LocalNumberField/5.11.0.1}{11} }^{4}$ R $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{4}$ $22^{2}$ R $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$ $22^{2}$ $22^{2}$

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
3Data not computed
7Data not computed
23Data not computed