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Properties

Label 20.0.31661970999...2281.2
Degree $20$
Signature $[0, 10]$
Discriminant $7^{10}\cdot 11^{18}\cdot 17^{10}$
Root discriminant $94.41$
Ramified primes $7, 11, 17$
Class number $145550$ (GRH)
Class group $[145550]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7316407296, 0, 5080838400, 0, 3223509696, 0, 727626240, 0, 127548036, 0, 13058815, 0, 1054867, 0, 55165, 0, 2288, 0, 55, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 55*x^18 + 2288*x^16 + 55165*x^14 + 1054867*x^12 + 13058815*x^10 + 127548036*x^8 + 727626240*x^6 + 3223509696*x^4 + 5080838400*x^2 + 7316407296)
 
gp: K = bnfinit(x^20 + 55*x^18 + 2288*x^16 + 55165*x^14 + 1054867*x^12 + 13058815*x^10 + 127548036*x^8 + 727626240*x^6 + 3223509696*x^4 + 5080838400*x^2 + 7316407296, 1)
 

Normalized defining polynomial

\( x^{20} + 55 x^{18} + 2288 x^{16} + 55165 x^{14} + 1054867 x^{12} + 13058815 x^{10} + 127548036 x^{8} + 727626240 x^{6} + 3223509696 x^{4} + 5080838400 x^{2} + 7316407296 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3166197099954132340458117662585349182281=7^{10}\cdot 11^{18}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $94.41$
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:  $7, 11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1309=7\cdot 11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1309}(1,·)$, $\chi_{1309}(834,·)$, $\chi_{1309}(645,·)$, $\chi_{1309}(545,·)$, $\chi_{1309}(713,·)$, $\chi_{1309}(783,·)$, $\chi_{1309}(1240,·)$, $\chi_{1309}(477,·)$, $\chi_{1309}(288,·)$, $\chi_{1309}(1121,·)$, $\chi_{1309}(356,·)$, $\chi_{1309}(1189,·)$, $\chi_{1309}(1191,·)$, $\chi_{1309}(426,·)$, $\chi_{1309}(1070,·)$, $\chi_{1309}(1072,·)$, $\chi_{1309}(50,·)$, $\chi_{1309}(307,·)$, $\chi_{1309}(1140,·)$, $\chi_{1309}(951,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{22} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{66} a^{11} - \frac{1}{6} a^{9} + \frac{1}{6} a^{7} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a$, $\frac{1}{396} a^{12} + \frac{1}{396} a^{10} - \frac{2}{9} a^{8} - \frac{7}{36} a^{6} - \frac{7}{36} a^{4} + \frac{11}{36} a^{2} - \frac{1}{2} a$, $\frac{1}{2376} a^{13} - \frac{17}{2376} a^{11} - \frac{11}{54} a^{9} - \frac{25}{216} a^{7} + \frac{65}{216} a^{5} - \frac{1}{2} a^{4} + \frac{101}{216} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{14256} a^{14} - \frac{17}{14256} a^{12} + \frac{17}{891} a^{10} - \frac{241}{1296} a^{8} - \frac{151}{1296} a^{6} + \frac{533}{1296} a^{4} - \frac{1}{2} a^{3} - \frac{5}{36} a^{2}$, $\frac{1}{171072} a^{15} - \frac{1}{28512} a^{14} + \frac{19}{171072} a^{13} + \frac{17}{28512} a^{12} + \frac{7}{3888} a^{11} + \frac{47}{3564} a^{10} - \frac{2473}{15552} a^{9} - \frac{407}{2592} a^{8} + \frac{893}{15552} a^{7} - \frac{497}{2592} a^{6} + \frac{5465}{15552} a^{5} - \frac{533}{2592} a^{4} - \frac{35}{72} a^{3} + \frac{23}{72} a^{2} - \frac{1}{12} a - \frac{1}{2}$, $\frac{1}{1026432} a^{16} - \frac{17}{1026432} a^{14} - \frac{1}{4752} a^{13} + \frac{115}{128304} a^{12} - \frac{19}{4752} a^{11} - \frac{13667}{1026432} a^{10} - \frac{7}{108} a^{9} - \frac{5983}{93312} a^{8} + \frac{97}{432} a^{7} + \frac{3125}{93312} a^{6} - \frac{29}{432} a^{5} - \frac{1175}{2592} a^{4} - \frac{65}{432} a^{3} - \frac{5}{18} a^{2} - \frac{1}{2}$, $\frac{1}{6158592} a^{17} - \frac{17}{6158592} a^{15} - \frac{1}{28512} a^{14} + \frac{115}{769824} a^{13} - \frac{19}{28512} a^{12} + \frac{2999}{559872} a^{11} - \frac{7}{648} a^{10} - \frac{5983}{559872} a^{9} - \frac{119}{2592} a^{8} + \frac{3125}{559872} a^{7} + \frac{403}{2592} a^{6} + \frac{121}{15552} a^{5} - \frac{281}{2592} a^{4} - \frac{41}{108} a^{3} + \frac{5}{12} a^{2} - \frac{5}{12} a$, $\frac{1}{3462368617687984799920820209152} a^{18} - \frac{1442629339178911116383933}{3462368617687984799920820209152} a^{16} - \frac{17788557664380307383837211}{865592154421996199980205052288} a^{14} - \frac{1}{4752} a^{13} - \frac{1096272084793522994361928787}{3462368617687984799920820209152} a^{12} + \frac{17}{4752} a^{11} - \frac{32344981970668298900911615505}{3462368617687984799920820209152} a^{10} - \frac{4}{27} a^{9} - \frac{58297607755104873168993454855}{314760783426180436356438200832} a^{8} + \frac{25}{432} a^{7} - \frac{233586517756443285926553097}{4371677547585839393839419456} a^{6} - \frac{65}{432} a^{5} + \frac{1517811452077696786785029}{4497610645664443820822448} a^{4} + \frac{115}{432} a^{3} + \frac{123611413143107754400773}{374800887138703651735204} a^{2} + \frac{5}{12} a - \frac{36576138451671063065019}{93700221784675912933801}$, $\frac{1}{20774211706127908799524921254912} a^{19} - \frac{1442629339178911116383933}{20774211706127908799524921254912} a^{17} + \frac{12570314193854688406714313}{5193552926531977199881230313728} a^{15} + \frac{1211002176432336685719987037}{20774211706127908799524921254912} a^{13} + \frac{5057148158677215913047862063}{20774211706127908799524921254912} a^{11} - \frac{201227176463675233350910029847}{1888564700557082618138629204992} a^{9} - \frac{3099126700375402055268055279}{26230065285515036363036516736} a^{7} - \frac{13143906699250425141416449}{1457225849195279797946473152} a^{5} - \frac{1}{2} a^{4} - \frac{48833434747028150733343}{3373207984248332865616836} a^{3} - \frac{118084314085789954977147}{374800887138703651735204} a - \frac{1}{2}$

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

Class group and class number

$C_{145550}$, which has order $145550$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1415140.16249 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{77}) \), \(\Q(\sqrt{-187}) \), \(\Q(\sqrt{-119}) \), \(\Q(\sqrt{77}, \sqrt{-119})\), \(\Q(\zeta_{11})^+\), 10.10.39630026842637.1, 10.0.3347948534700187.1, 10.0.5115361002064185719.3

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$7$7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed
$17$17.10.5.2$x^{10} - 83521 x^{2} + 8519142$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
17.10.5.2$x^{10} - 83521 x^{2} + 8519142$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$