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Properties

Label 44.44.1071632317...3536.1
Degree $44$
Signature $[44, 0]$
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![935089, 0, -316835961, 0, 6928067713, 0, -60026913481, 0, 274716257617, 0, -768135683609, 0, 1431629797153, 0, -1883097361001, 0, 1818601994737, 0, -1326378465209, 0, 745527993793, 0, -327608079881, 0, 113631570577, 0, -31279687001, 0, 6844037217, 0, -1187271081, 0, 162122289, 0, -17197497, 0, 1386817, 0, -82089, 0, 3361, 0, -85, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^44 - 85*x^42 + 3361*x^40 - 82089*x^38 + 1386817*x^36 - 17197497*x^34 + 162122289*x^32 - 1187271081*x^30 + 6844037217*x^28 - 31279687001*x^26 + 113631570577*x^24 - 327608079881*x^22 + 745527993793*x^20 - 1326378465209*x^18 + 1818601994737*x^16 - 1883097361001*x^14 + 1431629797153*x^12 - 768135683609*x^10 + 274716257617*x^8 - 60026913481*x^6 + 6928067713*x^4 - 316835961*x^2 + 935089)
 
gp: K = bnfinit(x^44 - 85*x^42 + 3361*x^40 - 82089*x^38 + 1386817*x^36 - 17197497*x^34 + 162122289*x^32 - 1187271081*x^30 + 6844037217*x^28 - 31279687001*x^26 + 113631570577*x^24 - 327608079881*x^22 + 745527993793*x^20 - 1326378465209*x^18 + 1818601994737*x^16 - 1883097361001*x^14 + 1431629797153*x^12 - 768135683609*x^10 + 274716257617*x^8 - 60026913481*x^6 + 6928067713*x^4 - 316835961*x^2 + 935089, 1)
 

Normalized defining polynomial

\( x^{44} - 85 x^{42} + 3361 x^{40} - 82089 x^{38} + 1386817 x^{36} - 17197497 x^{34} + 162122289 x^{32} - 1187271081 x^{30} + 6844037217 x^{28} - 31279687001 x^{26} + 113631570577 x^{24} - 327608079881 x^{22} + 745527993793 x^{20} - 1326378465209 x^{18} + 1818601994737 x^{16} - 1883097361001 x^{14} + 1431629797153 x^{12} - 768135683609 x^{10} + 274716257617 x^{8} - 60026913481 x^{6} + 6928067713 x^{4} - 316835961 x^{2} + 935089 \)

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

Invariants

Degree:  $44$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[44, 0]$
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$$\chi_{644}(1,·)$, $\chi_{644}(517,·)$, $\chi_{644}(321,·)$, $\chi_{644}(393,·)$, $\chi_{644}(139,·)$, $\chi_{644}(141,·)$, $\chi_{644}(15,·)$, $\chi_{644}(167,·)$, $\chi_{644}(531,·)$, $\chi_{644}(405,·)$, $\chi_{644}(279,·)$, $\chi_{644}(153,·)$, $\chi_{644}(27,·)$, $\chi_{644}(29,·)$, $\chi_{644}(155,·)$, $\chi_{644}(293,·)$, $\chi_{644}(295,·)$, $\chi_{644}(169,·)$, $\chi_{644}(43,·)$, $\chi_{644}(561,·)$, $\chi_{644}(307,·)$, $\chi_{644}(181,·)$, $\chi_{644}(183,·)$, $\chi_{644}(573,·)$, $\chi_{644}(449,·)$, $\chi_{644}(435,·)$, $\chi_{644}(267,·)$, $\chi_{644}(197,·)$, $\chi_{644}(97,·)$, $\chi_{644}(55,·)$, $\chi_{644}(335,·)$, $\chi_{644}(433,·)$, $\chi_{644}(85,·)$, $\chi_{644}(603,·)$, $\chi_{644}(223,·)$, $\chi_{644}(225,·)$, $\chi_{644}(587,·)$, $\chi_{644}(99,·)$, $\chi_{644}(363,·)$, $\chi_{644}(237,·)$, $\chi_{644}(631,·)$, $\chi_{644}(379,·)$, $\chi_{644}(125,·)$, $\chi_{644}(533,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{967} a^{23} - \frac{46}{967} a^{21} - \frac{47}{967} a^{19} + \frac{149}{967} a^{17} - \frac{354}{967} a^{15} - \frac{282}{967} a^{13} + \frac{180}{967} a^{11} + \frac{199}{967} a^{9} + \frac{148}{967} a^{7} - \frac{421}{967} a^{5} - \frac{168}{967} a^{3} + \frac{279}{967} a$, $\frac{1}{5025499} a^{24} + \frac{971789}{5025499} a^{22} + \frac{2469671}{5025499} a^{20} - \frac{1687266}{5025499} a^{18} + \frac{1150376}{5025499} a^{16} - \frac{818364}{5025499} a^{14} + \frac{1124801}{5025499} a^{12} - \frac{994844}{5025499} a^{10} + \frac{622896}{5025499} a^{8} + \frac{219088}{5025499} a^{6} - \frac{844359}{5025499} a^{4} - \frac{1784803}{5025499} a^{2} - \frac{456}{5197}$, $\frac{1}{5025499} a^{25} - \frac{50}{5025499} a^{23} + \frac{1944774}{5025499} a^{21} - \frac{1240324}{5025499} a^{19} + \frac{2085836}{5025499} a^{17} + \frac{1478710}{5025499} a^{15} - \frac{1219046}{5025499} a^{13} - \frac{33399}{5025499} a^{11} - \frac{1804103}{5025499} a^{9} + \frac{2126387}{5025499} a^{7} + \frac{1234441}{5025499} a^{5} + \frac{668181}{5025499} a^{3} - \frac{207087}{5025499} a$, $\frac{1}{5025499} a^{26} + \frac{279234}{5025499} a^{22} + \frac{1631250}{5025499} a^{20} - \frac{1869480}{5025499} a^{18} - \frac{1308478}{5025499} a^{16} - \frac{1933254}{5025499} a^{14} + \frac{926162}{5025499} a^{12} - \frac{1291313}{5025499} a^{10} - \frac{1907306}{5025499} a^{8} + \frac{2137843}{5025499} a^{6} - \frac{1345777}{5025499} a^{4} + \frac{1011745}{5025499} a^{2} - \frac{2012}{5197}$, $\frac{1}{5025499} a^{27} - \frac{1404}{5025499} a^{23} - \frac{535899}{5025499} a^{21} + \frac{1269508}{5025499} a^{19} + \frac{2105951}{5025499} a^{17} + \frac{1928117}{5025499} a^{15} - \frac{341906}{5025499} a^{13} - \frac{1551163}{5025499} a^{11} - \frac{2473779}{5025499} a^{9} + \frac{807411}{5025499} a^{7} + \frac{1216344}{5025499} a^{5} - \frac{2096061}{5025499} a^{3} + \frac{164378}{5025499} a$, $\frac{1}{5025499} a^{28} + \frac{1945628}{5025499} a^{22} + \frac{1093282}{5025499} a^{20} + \frac{194516}{5025499} a^{18} - \frac{1154657}{5025499} a^{16} + \frac{1514309}{5025499} a^{14} - \frac{337245}{5025499} a^{12} - \frac{2146033}{5025499} a^{10} + \frac{916569}{5025499} a^{8} + \frac{2260457}{5025499} a^{6} - \frac{1558333}{5025499} a^{4} + \frac{2024967}{5025499} a^{2} - \frac{993}{5197}$, $\frac{1}{5025499} a^{29} + \frac{1950}{5025499} a^{23} + \frac{43488}{5025499} a^{21} + \frac{1088400}{5025499} a^{19} + \frac{716263}{5025499} a^{17} + \frac{1082958}{5025499} a^{15} + \frac{560}{5025499} a^{13} - \frac{223143}{5025499} a^{11} + \frac{1088070}{5025499} a^{9} + \frac{1049556}{5025499} a^{7} - \frac{2426232}{5025499} a^{5} + \frac{1905436}{5025499} a^{3} - \frac{492501}{5025499} a$, $\frac{1}{5025499} a^{30} - \frac{331939}{5025499} a^{22} - \frac{342008}{5025499} a^{20} - \frac{816882}{5025499} a^{18} - \frac{777688}{5025499} a^{16} - \frac{2298322}{5025499} a^{14} - \frac{2467529}{5025499} a^{12} + \frac{1191256}{5025499} a^{10} - \frac{2452385}{5025499} a^{8} - \frac{2480417}{5025499} a^{6} + \frac{41814}{5025499} a^{4} + \frac{2228041}{5025499} a^{2} + \frac{513}{5197}$, $\frac{1}{5025499} a^{31} + \frac{669}{5025499} a^{23} - \frac{565479}{5025499} a^{21} - \frac{1372961}{5025499} a^{19} - \frac{1474086}{5025499} a^{17} + \frac{570422}{5025499} a^{15} - \frac{778504}{5025499} a^{13} + \frac{754708}{5025499} a^{11} - \frac{1594880}{5025499} a^{9} + \frac{1516076}{5025499} a^{7} + \frac{727818}{5025499} a^{5} + \frac{1630386}{5025499} a^{3} - \frac{2190778}{5025499} a$, $\frac{1}{5025499} a^{32} - \frac{2402949}{5025499} a^{22} - \frac{193689}{5025499} a^{20} + \frac{1595092}{5025499} a^{18} - \frac{129775}{5025499} a^{16} - \frac{1072379}{5025499} a^{14} + \frac{2087689}{5025499} a^{12} + \frac{589888}{5025499} a^{10} + \frac{1915069}{5025499} a^{8} - \frac{102583}{5025499} a^{6} - \frac{1374830}{5025499} a^{4} + \frac{799166}{5025499} a^{2} - \frac{1559}{5197}$, $\frac{1}{5025499} a^{33} - \frac{1935}{5025499} a^{23} - \frac{79355}{5025499} a^{21} - \frac{691588}{5025499} a^{19} + \frac{810882}{5025499} a^{17} - \frac{1722004}{5025499} a^{15} - \frac{1581393}{5025499} a^{13} + \frac{579494}{5025499} a^{11} + \frac{2294450}{5025499} a^{9} - \frac{1562940}{5025499} a^{7} - \frac{2076425}{5025499} a^{5} - \frac{531266}{5025499} a^{3} - \frac{16014}{5025499} a$, $\frac{1}{5025499} a^{34} + \frac{795734}{5025499} a^{22} - \frac{1127752}{5025499} a^{20} - \frac{2499977}{5025499} a^{18} - \frac{2040501}{5025499} a^{16} - \frac{2083548}{5025499} a^{14} + \frac{1028362}{5025499} a^{12} + \frac{2037427}{5025499} a^{10} - \frac{2378940}{5025499} a^{8} - \frac{283061}{5025499} a^{6} - \frac{1078756}{5025499} a^{4} - \frac{1092006}{5025499} a^{2} + \frac{1130}{5197}$, $\frac{1}{5025499} a^{35} + \frac{593}{5025499} a^{23} + \frac{270241}{5025499} a^{21} - \frac{306843}{5025499} a^{19} + \frac{95466}{5025499} a^{17} - \frac{2031578}{5025499} a^{15} - \frac{889331}{5025499} a^{13} - \frac{373981}{5025499} a^{11} + \frac{203969}{5025499} a^{9} - \frac{2377452}{5025499} a^{7} + \frac{1992671}{5025499} a^{5} + \frac{1828708}{5025499} a^{3} + \frac{370327}{5025499} a$, $\frac{1}{5025499} a^{36} + \frac{1931749}{5025499} a^{22} - \frac{2401537}{5025499} a^{20} + \frac{569903}{5025499} a^{18} - \frac{736682}{5025499} a^{16} + \frac{1952617}{5025499} a^{14} + \frac{1010393}{5025499} a^{12} + \frac{2163078}{5025499} a^{10} + \frac{132146}{5025499} a^{8} - \frac{2289038}{5025499} a^{6} - \frac{16305}{5025499} a^{4} - \frac{1621783}{5025499} a^{2} + \frac{164}{5197}$, $\frac{1}{5025499} a^{37} - \frac{1535}{5025499} a^{23} + \frac{1096044}{5025499} a^{21} + \frac{975269}{5025499} a^{19} - \frac{2342555}{5025499} a^{17} - \frac{2158210}{5025499} a^{15} - \frac{1582910}{5025499} a^{13} + \frac{931389}{5025499} a^{11} + \frac{2372053}{5025499} a^{9} - \frac{1961627}{5025499} a^{7} - \frac{234579}{5025499} a^{5} + \frac{1537993}{5025499} a^{3} - \frac{1499255}{5025499} a$, $\frac{1}{5025499} a^{38} + \frac{218956}{5025499} a^{22} - \frac{2331491}{5025499} a^{20} + \frac{861619}{5025499} a^{18} - \frac{281199}{5025499} a^{16} - \frac{1396900}{5025499} a^{14} - \frac{1270732}{5025499} a^{12} - \frac{1987290}{5025499} a^{10} - \frac{661077}{5025499} a^{8} - \frac{642932}{5025499} a^{6} + \frac{2025670}{5025499} a^{4} - \frac{2274905}{5025499} a^{2} + \frac{1635}{5197}$, $\frac{1}{5025499} a^{39} + \frac{682}{5025499} a^{23} - \frac{2341885}{5025499} a^{21} + \frac{1069499}{5025499} a^{19} + \frac{2374468}{5025499} a^{17} + \frac{489611}{5025499} a^{15} - \frac{23452}{5025499} a^{13} - \frac{1072618}{5025499} a^{11} + \frac{1131888}{5025499} a^{9} + \frac{2231009}{5025499} a^{7} - \frac{1565457}{5025499} a^{5} - \frac{783366}{5025499} a^{3} + \frac{988587}{5025499} a$, $\frac{1}{5025499} a^{40} - \frac{1736115}{5025499} a^{22} + \frac{296042}{5025499} a^{20} + \frac{2250609}{5025499} a^{18} - \frac{88977}{5025499} a^{16} + \frac{270407}{5025499} a^{14} + \frac{714447}{5025499} a^{12} + \frac{1173131}{5025499} a^{10} - \frac{442147}{5025499} a^{8} - \frac{218503}{5025499} a^{6} + \frac{2162586}{5025499} a^{4} + \frac{2053475}{5025499} a^{2} - \frac{828}{5197}$, $\frac{1}{5025499} a^{41} - \frac{317}{5025499} a^{23} + \frac{857318}{5025499} a^{21} + \frac{1076087}{5025499} a^{19} + \frac{2244476}{5025499} a^{17} - \frac{1091207}{5025499} a^{15} - \frac{1307186}{5025499} a^{13} + \frac{2035833}{5025499} a^{11} - \frac{1777776}{5025499} a^{9} + \frac{379152}{5025499} a^{7} + \frac{88983}{5025499} a^{5} + \frac{1918353}{5025499} a^{3} + \frac{1039062}{5025499} a$, $\frac{1}{5025499} a^{42} + \frac{2358992}{5025499} a^{22} - \frac{16050}{5025499} a^{20} + \frac{84048}{5025499} a^{18} + \frac{1742057}{5025499} a^{16} + \frac{597374}{5025499} a^{14} + \frac{1787321}{5025499} a^{12} - \frac{536887}{5025499} a^{10} + \frac{1842723}{5025499} a^{8} - \frac{817107}{5025499} a^{6} + \frac{607997}{5025499} a^{4} - \frac{1887601}{5025499} a^{2} + \frac{964}{5197}$, $\frac{1}{5025499} a^{43} - \frac{446}{5025499} a^{23} - \frac{2042880}{5025499} a^{21} + \frac{416656}{5025499} a^{19} + \frac{1970725}{5025499} a^{17} + \frac{1605592}{5025499} a^{15} - \frac{1242530}{5025499} a^{13} + \frac{1931688}{5025499} a^{11} - \frac{314032}{5025499} a^{9} + \frac{1770999}{5025499} a^{7} - \frac{1117407}{5025499} a^{5} + \frac{2509061}{5025499} a^{3} + \frac{989355}{5025499} a$

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:  $43$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
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

\(\Q(\sqrt{161}) \), \(\Q(\sqrt{23}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{7}, \sqrt{23})\), \(\Q(\zeta_{23})^+\), 22.22.78048218870425324004237696277333187889.1, \(\Q(\zeta_{92})^+\), 22.22.14232954634830452964011747236817552143286272.1

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 $22^{2}$ $22^{2}$ R $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{4}$ 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