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Properties

Label 20.20.7428460430...7232.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{40}\cdot 3^{10}\cdot 10273^{5}$
Root discriminant $69.75$
Ramified primes $2, 3, 10273$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times D_5\wr C_2$ (as 20T92)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![832113, 0, -2981448, 0, 4502727, 0, -3766752, 0, 1928971, 0, -630384, 0, 132675, 0, -17660, 0, 1410, 0, -60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 60*x^18 + 1410*x^16 - 17660*x^14 + 132675*x^12 - 630384*x^10 + 1928971*x^8 - 3766752*x^6 + 4502727*x^4 - 2981448*x^2 + 832113)
 
gp: K = bnfinit(x^20 - 60*x^18 + 1410*x^16 - 17660*x^14 + 132675*x^12 - 630384*x^10 + 1928971*x^8 - 3766752*x^6 + 4502727*x^4 - 2981448*x^2 + 832113, 1)
 

Normalized defining polynomial

\( x^{20} - 60 x^{18} + 1410 x^{16} - 17660 x^{14} + 132675 x^{12} - 630384 x^{10} + 1928971 x^{8} - 3766752 x^{6} + 4502727 x^{4} - 2981448 x^{2} + 832113 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7428460430560760661253303852421087232=2^{40}\cdot 3^{10}\cdot 10273^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.75$
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:  $2, 3, 10273$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{10} + \frac{2}{9} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} + \frac{2}{9} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{27} a^{13} - \frac{1}{27} a^{11} - \frac{1}{27} a^{7} + \frac{10}{27} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{27} a^{14} - \frac{1}{27} a^{12} - \frac{1}{27} a^{8} + \frac{1}{27} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{27} a^{15} - \frac{1}{27} a^{11} - \frac{1}{27} a^{9} - \frac{8}{27} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{459} a^{16} + \frac{2}{153} a^{14} - \frac{25}{459} a^{12} - \frac{19}{459} a^{10} + \frac{13}{153} a^{8} + \frac{7}{459} a^{6} + \frac{6}{17} a^{4} - \frac{5}{51} a^{2} - \frac{3}{17}$, $\frac{1}{459} a^{17} + \frac{2}{153} a^{15} - \frac{8}{459} a^{13} + \frac{5}{153} a^{11} + \frac{13}{153} a^{9} - \frac{10}{459} a^{7} - \frac{25}{459} a^{5} + \frac{4}{17} a^{3} + \frac{8}{51} a$, $\frac{1}{161383579767} a^{18} - \frac{168532963}{161383579767} a^{16} + \frac{289795759}{17931508863} a^{14} + \frac{3512721821}{161383579767} a^{12} + \frac{5365633609}{161383579767} a^{10} + \frac{2719881157}{17931508863} a^{8} - \frac{5715100244}{53794526589} a^{6} + \frac{9850853540}{53794526589} a^{4} + \frac{3727515349}{17931508863} a^{2} - \frac{231677136}{5977169621}$, $\frac{1}{484150739301} a^{19} + \frac{61021750}{161383579767} a^{17} + \frac{1572583703}{161383579767} a^{15} - \frac{5277233504}{484150739301} a^{13} + \frac{5538925475}{161383579767} a^{11} - \frac{23132597486}{161383579767} a^{9} - \frac{14684113241}{484150739301} a^{7} + \frac{4532388476}{17931508863} a^{5} - \frac{9984814958}{53794526589} a^{3} - \frac{428823925}{5977169621} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $19$
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:  \( 1609466316240 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_5\wr C_2$ (as 20T92):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $C_2\times D_5\wr C_2$
Character table for $C_2\times D_5\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.23668992.2, 10.10.26260367920128.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
10273Data not computed