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Properties

Label 20.0.29133682277...7536.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 11^{16}$
Root discriminant $33.36$
Ramified primes $2, 3, 11$
Class number $5$ (GRH)
Class group $[5]$ (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![1024, 0, -7680, 0, 48640, 0, -60032, 0, 50944, 0, -22752, 0, 7264, 0, -1456, 0, 212, 0, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 18*x^18 + 212*x^16 - 1456*x^14 + 7264*x^12 - 22752*x^10 + 50944*x^8 - 60032*x^6 + 48640*x^4 - 7680*x^2 + 1024)
 
gp: K = bnfinit(x^20 - 18*x^18 + 212*x^16 - 1456*x^14 + 7264*x^12 - 22752*x^10 + 50944*x^8 - 60032*x^6 + 48640*x^4 - 7680*x^2 + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - 18 x^{18} + 212 x^{16} - 1456 x^{14} + 7264 x^{12} - 22752 x^{10} + 50944 x^{8} - 60032 x^{6} + 48640 x^{4} - 7680 x^{2} + 1024 \)

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:  \(2913368227796180298080018497536=2^{30}\cdot 3^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.36$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(264=2^{3}\cdot 3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{264}(1,·)$, $\chi_{264}(67,·)$, $\chi_{264}(179,·)$, $\chi_{264}(257,·)$, $\chi_{264}(137,·)$, $\chi_{264}(203,·)$, $\chi_{264}(163,·)$, $\chi_{264}(25,·)$, $\chi_{264}(89,·)$, $\chi_{264}(91,·)$, $\chi_{264}(97,·)$, $\chi_{264}(251,·)$, $\chi_{264}(113,·)$, $\chi_{264}(169,·)$, $\chi_{264}(235,·)$, $\chi_{264}(49,·)$, $\chi_{264}(115,·)$, $\chi_{264}(155,·)$, $\chi_{264}(185,·)$, $\chi_{264}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{897049359872} a^{18} + \frac{335285807}{448524679936} a^{16} + \frac{44712727}{224262339968} a^{14} - \frac{57936275}{112131169984} a^{12} + \frac{94779135}{28032792496} a^{10} - \frac{205566055}{28032792496} a^{8} - \frac{35266913}{7008198124} a^{6} - \frac{144739027}{3504099062} a^{4} - \frac{326033502}{1752049531} a^{2} - \frac{96297369}{1752049531}$, $\frac{1}{897049359872} a^{19} + \frac{335285807}{448524679936} a^{17} + \frac{44712727}{224262339968} a^{15} - \frac{57936275}{112131169984} a^{13} + \frac{94779135}{28032792496} a^{11} - \frac{205566055}{28032792496} a^{9} - \frac{35266913}{7008198124} a^{7} - \frac{144739027}{3504099062} a^{5} - \frac{326033502}{1752049531} a^{3} - \frac{96297369}{1752049531} a$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( -\frac{153716985}{897049359872} a^{18} + \frac{170773799}{56065584992} a^{16} - \frac{8000903505}{224262339968} a^{14} + \frac{27138383201}{112131169984} a^{12} - \frac{33543167157}{28032792496} a^{10} + \frac{51429095339}{14016396248} a^{8} - \frac{113467761887}{14016396248} a^{6} + \frac{63026312027}{7008198124} a^{4} - \frac{13340351098}{1752049531} a^{2} + \frac{2105602296}{1752049531} \) (order $6$)
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:  \( 5268231.17547 \) (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

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\zeta_{11})^+\), 10.0.7024111812608.1, 10.0.52089208083.1, 10.10.1706859170463744.1

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 R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$