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Properties

Label 44.0.10716323175...3536.1
Degree $44$
Signature $[0, 22]$
Discriminant $2^{44}\cdot 7^{22}\cdot 23^{42}$
Root discriminant $105.54$
Ramified primes $2, 7, 23$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{22}$ (as 44T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27008809, 0, 2191778007, 0, 46621531945, 0, 380494930135, 0, 1602545786665, 0, 4049120256215, 0, 6740424007465, 0, 7872887316695, 0, 6740424007465, 0, 4362827243735, 0, 2182247178025, 0, 856537374935, 0, 266491698985, 0, 66097680599, 0, 13090266921, 0, 2064667863, 0, 257459241, 0, 25045527, 0, 1859689, 0, 101751, 0, 3865, 0, 91, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^44 + 91*x^42 + 3865*x^40 + 101751*x^38 + 1859689*x^36 + 25045527*x^34 + 257459241*x^32 + 2064667863*x^30 + 13090266921*x^28 + 66097680599*x^26 + 266491698985*x^24 + 856537374935*x^22 + 2182247178025*x^20 + 4362827243735*x^18 + 6740424007465*x^16 + 7872887316695*x^14 + 6740424007465*x^12 + 4049120256215*x^10 + 1602545786665*x^8 + 380494930135*x^6 + 46621531945*x^4 + 2191778007*x^2 + 27008809)
 
gp: K = bnfinit(x^44 + 91*x^42 + 3865*x^40 + 101751*x^38 + 1859689*x^36 + 25045527*x^34 + 257459241*x^32 + 2064667863*x^30 + 13090266921*x^28 + 66097680599*x^26 + 266491698985*x^24 + 856537374935*x^22 + 2182247178025*x^20 + 4362827243735*x^18 + 6740424007465*x^16 + 7872887316695*x^14 + 6740424007465*x^12 + 4049120256215*x^10 + 1602545786665*x^8 + 380494930135*x^6 + 46621531945*x^4 + 2191778007*x^2 + 27008809, 1)
 

Normalized defining polynomial

\( x^{44} + 91 x^{42} + 3865 x^{40} + 101751 x^{38} + 1859689 x^{36} + 25045527 x^{34} + 257459241 x^{32} + 2064667863 x^{30} + 13090266921 x^{28} + 66097680599 x^{26} + 266491698985 x^{24} + 856537374935 x^{22} + 2182247178025 x^{20} + 4362827243735 x^{18} + 6740424007465 x^{16} + 7872887316695 x^{14} + 6740424007465 x^{12} + 4049120256215 x^{10} + 1602545786665 x^{8} + 380494930135 x^{6} + 46621531945 x^{4} + 2191778007 x^{2} + 27008809 \)

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

Invariants

Degree:  $44$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 22]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(107163231750047944848560960956807209663182497699341984637548299223802110981295940395073536=2^{44}\cdot 7^{22}\cdot 23^{42}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.54$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 7, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(644=2^{2}\cdot 7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{644}(1,·)$, $\chi_{644}(643,·)$, $\chi_{644}(517,·)$, $\chi_{644}(519,·)$, $\chi_{644}(393,·)$, $\chi_{644}(141,·)$, $\chi_{644}(195,·)$, $\chi_{644}(405,·)$, $\chi_{644}(407,·)$, $\chi_{644}(127,·)$, $\chi_{644}(153,·)$, $\chi_{644}(111,·)$, $\chi_{644}(29,·)$, $\chi_{644}(419,·)$, $\chi_{644}(293,·)$, $\chi_{644}(561,·)$, $\chi_{644}(169,·)$, $\chi_{644}(615,·)$, $\chi_{644}(321,·)$, $\chi_{644}(559,·)$, $\chi_{644}(433,·)$, $\chi_{644}(181,·)$, $\chi_{644}(573,·)$, $\chi_{644}(447,·)$, $\chi_{644}(449,·)$, $\chi_{644}(323,·)$, $\chi_{644}(197,·)$, $\chi_{644}(71,·)$, $\chi_{644}(463,·)$, $\chi_{644}(83,·)$, $\chi_{644}(85,·)$, $\chi_{644}(475,·)$, $\chi_{644}(351,·)$, $\chi_{644}(225,·)$, $\chi_{644}(97,·)$, $\chi_{644}(491,·)$, $\chi_{644}(237,·)$, $\chi_{644}(239,·)$, $\chi_{644}(211,·)$, $\chi_{644}(547,·)$, $\chi_{644}(503,·)$, $\chi_{644}(251,·)$, $\chi_{644}(125,·)$, $\chi_{644}(533,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{5197} a^{23} + \frac{46}{5197} a^{21} + \frac{920}{5197} a^{19} + \frac{94}{5197} a^{17} + \frac{2314}{5197} a^{15} + \frac{2137}{5197} a^{13} + \frac{1034}{5197} a^{11} + \frac{175}{5197} a^{9} + \frac{210}{5197} a^{7} + \frac{140}{5197} a^{5} - \frac{1556}{5197} a^{3} + \frac{331}{5197} a$, $\frac{1}{5025499} a^{24} + \frac{971885}{5025499} a^{22} - \frac{2467655}{5025499} a^{20} - \frac{1662946}{5025499} a^{18} - \frac{964328}{5025499} a^{16} + \frac{121668}{5025499} a^{14} + \frac{2043455}{5025499} a^{12} + \frac{1008393}{5025499} a^{10} - \frac{789734}{5025499} a^{8} - \frac{1631718}{5025499} a^{6} - \frac{666772}{5025499} a^{4} - \frac{1194979}{5025499} a^{2} + \frac{88}{967}$, $\frac{1}{5025499} a^{25} + \frac{50}{5025499} a^{23} - \frac{1942574}{5025499} a^{21} - \frac{1212324}{5025499} a^{19} - \frac{1857836}{5025499} a^{17} - \frac{2306469}{5025499} a^{15} + \frac{763147}{5025499} a^{13} + \frac{1230803}{5025499} a^{11} + \frac{6107}{5025499} a^{9} + \frac{328391}{5025499} a^{7} - \frac{1035199}{5025499} a^{5} - \frac{1694918}{5025499} a^{3} + \frac{411887}{5025499} a$, $\frac{1}{5025499} a^{26} - \frac{281834}{5025499} a^{22} + \frac{1558450}{5025499} a^{20} + \frac{881480}{5025499} a^{18} + \frac{680440}{5025499} a^{16} - \frac{294754}{5025499} a^{14} - \frac{431967}{5025499} a^{12} - \frac{158553}{5025499} a^{10} - \frac{388901}{5025499} a^{8} + \frac{142717}{5025499} a^{6} + \frac{1490688}{5025499} a^{4} - \frac{145151}{5025499} a^{2} + \frac{435}{967}$, $\frac{1}{5025499} a^{27} - \frac{437}{5025499} a^{23} - \frac{573785}{5025499} a^{21} - \frac{1559228}{5025499} a^{19} + \frac{2004263}{5025499} a^{17} - \frac{2456966}{5025499} a^{15} - \frac{2146458}{5025499} a^{13} - \frac{672997}{5025499} a^{11} - \frac{1399416}{5025499} a^{9} - \frac{1069901}{5025499} a^{7} + \frac{682276}{5025499} a^{5} - \frac{780470}{5025499} a^{3} - \frac{81379}{5025499} a$, $\frac{1}{5025499} a^{28} + \frac{1998044}{5025499} a^{22} + \frac{557822}{5025499} a^{20} - \frac{1031283}{5025499} a^{18} - \frac{1726386}{5025499} a^{16} + \frac{767468}{5025499} a^{14} - \frac{2221984}{5025499} a^{12} + \frac{2049912}{5025499} a^{10} + \frac{575772}{5025499} a^{8} + \frac{1242368}{5025499} a^{6} - \frac{680892}{5025499} a^{4} + \frac{364694}{5025499} a^{2} - \frac{224}{967}$, $\frac{1}{5025499} a^{29} + \frac{222}{5025499} a^{23} - \frac{883008}{5025499} a^{21} + \frac{305111}{5025499} a^{19} + \frac{1447308}{5025499} a^{17} + \frac{1266440}{5025499} a^{15} + \frac{106552}{5025499} a^{13} + \frac{1782053}{5025499} a^{11} - \frac{2283647}{5025499} a^{9} - \frac{1183835}{5025499} a^{7} + \frac{1051972}{5025499} a^{5} - \frac{1808155}{5025499} a^{3} + \frac{922658}{5025499} a$, $\frac{1}{5025499} a^{30} - \frac{545021}{5025499} a^{22} + \frac{345130}{5025499} a^{20} - \frac{1265606}{5025499} a^{18} - \frac{749201}{5025499} a^{16} - \frac{1776249}{5025499} a^{14} + \frac{429953}{5025499} a^{12} + \frac{562}{5025499} a^{10} - \frac{1755352}{5025499} a^{8} + \frac{1457440}{5025499} a^{6} + \frac{475758}{5025499} a^{4} - \frac{143451}{5025499} a^{2} - \frac{196}{967}$, $\frac{1}{5025499} a^{31} + \frac{367}{5025499} a^{23} + \frac{305483}{5025499} a^{21} - \frac{2058546}{5025499} a^{19} + \frac{262281}{5025499} a^{17} - \frac{1148666}{5025499} a^{15} + \frac{8341}{5025499} a^{13} + \frac{1075866}{5025499} a^{11} - \frac{1796933}{5025499} a^{9} + \frac{402443}{5025499} a^{7} + \frac{1447593}{5025499} a^{5} + \frac{542152}{5025499} a^{3} - \frac{1413148}{5025499} a$, $\frac{1}{5025499} a^{32} + \frac{434117}{5025499} a^{22} - \frac{1018981}{5025499} a^{20} + \frac{2478084}{5025499} a^{18} + \frac{974780}{5025499} a^{16} + \frac{585676}{5025499} a^{14} - \frac{72768}{5025499} a^{12} + \frac{9762}{5025499} a^{10} - \frac{1244121}{5025499} a^{8} + \frac{2253718}{5025499} a^{6} - \frac{1001975}{5025499} a^{4} - \frac{74268}{5025499} a^{2} - \frac{385}{967}$, $\frac{1}{5025499} a^{33} - \frac{66}{5025499} a^{23} - \frac{889403}{5025499} a^{21} + \frac{44145}{5025499} a^{19} + \frac{365570}{5025499} a^{17} + \frac{986014}{5025499} a^{15} + \frac{1795476}{5025499} a^{13} - \frac{1666049}{5025499} a^{11} - \frac{1843661}{5025499} a^{9} + \frac{1534270}{5025499} a^{7} - \frac{1481607}{5025499} a^{5} + \frac{2097614}{5025499} a^{3} + \frac{24053}{5025499} a$, $\frac{1}{5025499} a^{34} - \frac{2076480}{5025499} a^{22} - \frac{2005117}{5025499} a^{20} + \frac{1172112}{5025499} a^{18} - \frac{2353646}{5025499} a^{16} - \frac{225434}{5025499} a^{14} - \frac{2486492}{5025499} a^{12} - \frac{621210}{5025499} a^{10} - \frac{333184}{5025499} a^{8} + \frac{1385983}{5025499} a^{6} - \frac{1705346}{5025499} a^{4} + \frac{1563423}{5025499} a^{2} + \frac{6}{967}$, $\frac{1}{5025499} a^{35} - \frac{331}{5025499} a^{23} - \frac{1986744}{5025499} a^{21} + \frac{1539572}{5025499} a^{19} + \frac{1835398}{5025499} a^{17} - \frac{393692}{5025499} a^{15} + \frac{1753803}{5025499} a^{13} + \frac{228783}{5025499} a^{11} + \frac{1156963}{5025499} a^{9} + \frac{158860}{5025499} a^{7} + \frac{2502071}{5025499} a^{5} + \frac{2471436}{5025499} a^{3} - \frac{1256862}{5025499} a$, $\frac{1}{5025499} a^{36} - \frac{1924745}{5025499} a^{22} - \frac{1123395}{5025499} a^{20} - \frac{820337}{5025499} a^{18} + \frac{2045676}{5025499} a^{16} + \frac{1821919}{5025499} a^{14} - \frac{1829977}{5025499} a^{12} - \frac{1773387}{5025499} a^{10} + \frac{82854}{5025499} a^{8} + \frac{131806}{5025499} a^{6} - \frac{2133639}{5025499} a^{4} + \frac{219510}{5025499} a^{2} + \frac{118}{967}$, $\frac{1}{5025499} a^{37} - \frac{415}{5025499} a^{23} + \frac{1962302}{5025499} a^{21} + \frac{587615}{5025499} a^{19} + \frac{2014732}{5025499} a^{17} + \frac{2129425}{5025499} a^{15} - \frac{394949}{5025499} a^{13} - \frac{2113771}{5025499} a^{11} + \frac{132171}{5025499} a^{9} + \frac{2201186}{5025499} a^{7} + \frac{921114}{5025499} a^{5} + \frac{1159434}{5025499} a^{3} - \frac{671897}{5025499} a$, $\frac{1}{5025499} a^{38} - \frac{1770842}{5025499} a^{22} + \frac{1712586}{5025499} a^{20} + \frac{385505}{5025499} a^{18} - \frac{1052274}{5025499} a^{16} - \frac{157719}{5025499} a^{14} + \frac{1636222}{5025499} a^{12} + \frac{1498849}{5025499} a^{10} + \frac{1119011}{5025499} a^{8} + \frac{2200509}{5025499} a^{6} + \frac{851499}{5025499} a^{4} + \frac{936219}{5025499} a^{2} - \frac{226}{967}$, $\frac{1}{5025499} a^{39} - \frac{265}{5025499} a^{23} - \frac{2274355}{5025499} a^{21} + \frac{1054669}{5025499} a^{19} - \frac{459503}{5025499} a^{17} + \frac{1175774}{5025499} a^{15} + \frac{1158524}{5025499} a^{13} - \frac{2031668}{5025499} a^{11} - \frac{610952}{5025499} a^{9} + \frac{2134753}{5025499} a^{7} + \frac{2482828}{5025499} a^{5} - \frac{108141}{5025499} a^{3} + \frac{1928581}{5025499} a$, $\frac{1}{5025499} a^{40} - \frac{1025279}{5025499} a^{22} + \frac{440964}{5025499} a^{20} + \frac{1103719}{5025499} a^{18} + \frac{1929303}{5025499} a^{16} - \frac{1777949}{5025499} a^{14} + \frac{1755514}{5025499} a^{12} + \frac{261746}{5025499} a^{10} - \frac{1099298}{5025499} a^{8} + \frac{2270472}{5025499} a^{6} - \frac{910256}{5025499} a^{4} + \frac{1865583}{5025499} a^{2} + \frac{112}{967}$, $\frac{1}{5025499} a^{41} - \frac{259}{5025499} a^{23} + \frac{2362393}{5025499} a^{21} - \frac{671693}{5025499} a^{19} - \frac{2228797}{5025499} a^{17} - \frac{1917197}{5025499} a^{15} + \frac{1105690}{5025499} a^{13} - \frac{247863}{5025499} a^{11} + \frac{2386737}{5025499} a^{9} + \frac{1428215}{5025499} a^{7} + \frac{1878572}{5025499} a^{5} + \frac{17646}{5025499} a^{3} - \frac{1870248}{5025499} a$, $\frac{1}{5025499} a^{42} - \frac{2219841}{5025499} a^{22} - \frac{1555965}{5025499} a^{20} - \frac{738897}{5025499} a^{18} - \frac{403199}{5025499} a^{16} + \frac{2464708}{5025499} a^{14} + \frac{1329587}{5025499} a^{12} + \frac{2234576}{5025499} a^{10} - \frac{2092931}{5025499} a^{8} + \frac{1405526}{5025499} a^{6} - \frac{1809336}{5025499} a^{4} + \frac{211129}{5025499} a^{2} - \frac{416}{967}$, $\frac{1}{5025499} a^{43} + \frac{391}{5025499} a^{23} + \frac{64727}{5025499} a^{21} + \frac{1521949}{5025499} a^{19} + \frac{2253150}{5025499} a^{17} - \frac{1003921}{5025499} a^{15} + \frac{1894315}{5025499} a^{13} + \frac{1301421}{5025499} a^{11} - \frac{515754}{5025499} a^{9} + \frac{282839}{5025499} a^{7} + \frac{2467705}{5025499} a^{5} - \frac{1952050}{5025499} a^{3} - \frac{988014}{5025499} a$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $21$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1}{5197} a^{23} - \frac{46}{5197} a^{21} - \frac{920}{5197} a^{19} - \frac{10488}{5197} a^{17} - \frac{75072}{5197} a^{15} - \frac{350336}{5197} a^{13} - \frac{1071616}{5197} a^{11} - \frac{2104960}{5197} a^{9} - \frac{2525952}{5197} a^{7} - \frac{1683968}{5197} a^{5} - \frac{518144}{5197} a^{3} - \frac{47104}{5197} a \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{161}) \), \(\Q(\sqrt{-161}) \), \(\Q(i, \sqrt{161})\), \(\Q(\zeta_{23})^+\), 22.0.7198079267989980836471065337135104.1, 22.22.78048218870425324004237696277333187889.1, 22.0.327357956601100418172270186446803699295584256.1

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
7Data not computed
23Data not computed