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Properties

Label 20.0.34554763942...8064.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 29^{4}\cdot 53^{4}$
Root discriminant $15.03$
Ramified primes $2, 3, 29, 53$
Class number $1$
Class group Trivial
Galois group $C_2\times D_5\wr C_2$ (as 20T100)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 4, 0, -1, 2, 4, -8, -1, -6, -1, -4, 14, 8, -6, -2, 0, 0, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^18 - 2*x^15 - 6*x^14 + 8*x^13 + 14*x^12 - 4*x^11 - x^10 - 6*x^9 - x^8 - 8*x^7 + 4*x^6 + 2*x^5 - x^4 + 4*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^18 - 2*x^15 - 6*x^14 + 8*x^13 + 14*x^12 - 4*x^11 - x^10 - 6*x^9 - x^8 - 8*x^7 + 4*x^6 + 2*x^5 - x^4 + 4*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{18} - 2 x^{15} - 6 x^{14} + 8 x^{13} + 14 x^{12} - 4 x^{11} - x^{10} - 6 x^{9} - x^{8} - 8 x^{7} + 4 x^{6} + 2 x^{5} - x^{4} + 4 x^{2} - 2 x + 1 \)

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:  \(345547639425403337048064=2^{20}\cdot 3^{10}\cdot 29^{4}\cdot 53^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.03$
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:  $2, 3, 29, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{5} a^{17} - \frac{2}{5} a^{15} - \frac{1}{5} a^{14} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{18} - \frac{2}{5} a^{16} - \frac{1}{5} a^{15} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{291635} a^{19} - \frac{4311}{291635} a^{18} - \frac{21594}{291635} a^{17} + \frac{118496}{291635} a^{16} - \frac{1678}{58327} a^{15} + \frac{123202}{291635} a^{14} - \frac{22964}{58327} a^{13} + \frac{84433}{291635} a^{12} - \frac{30169}{291635} a^{11} - \frac{127309}{291635} a^{10} + \frac{88682}{291635} a^{9} + \frac{25377}{291635} a^{8} + \frac{79531}{291635} a^{7} - \frac{128697}{291635} a^{6} + \frac{6347}{291635} a^{5} - \frac{6552}{291635} a^{4} - \frac{588}{3995} a^{3} + \frac{88}{799} a^{2} - \frac{117672}{291635} a + \frac{72398}{291635}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{4422086}{291635} a^{19} - \frac{2958611}{291635} a^{18} + \frac{8642814}{291635} a^{17} + \frac{6882378}{291635} a^{16} + \frac{1013342}{291635} a^{15} + \frac{6919338}{291635} a^{14} + \frac{6200391}{58327} a^{13} - \frac{17402063}{291635} a^{12} - \frac{85892062}{291635} a^{11} - \frac{32004044}{291635} a^{10} + \frac{17552176}{291635} a^{9} + \frac{49058121}{291635} a^{8} + \frac{28803924}{291635} a^{7} + \frac{35527126}{291635} a^{6} - \frac{4460294}{291635} a^{5} - \frac{25452061}{291635} a^{4} - \frac{138158}{3995} a^{3} + \frac{48508}{3995} a^{2} - \frac{2378604}{58327} a - \frac{861738}{291635} \) (order $12$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 17343.1636986 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_5\wr C_2$ (as 20T100):

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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{12})\), 10.0.2419065856.1

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 ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.10.0.1$x^{10} + x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$53$53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.10.0.1$x^{10} - x + 19$$1$$10$$0$$C_{10}$$[\ ]^{10}$