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Properties

Label 20.4.35201711859...3104.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{46}\cdot 29^{8}$
Root discriminant $18.94$
Ramified primes $2, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T230

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 0, -280, 0, 996, 0, -1472, 0, 1112, 0, -600, 0, 328, 0, -128, 0, 44, 0, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 + 44*x^16 - 128*x^14 + 328*x^12 - 600*x^10 + 1112*x^8 - 1472*x^6 + 996*x^4 - 280*x^2 + 16)
 
gp: K = bnfinit(x^20 - 10*x^18 + 44*x^16 - 128*x^14 + 328*x^12 - 600*x^10 + 1112*x^8 - 1472*x^6 + 996*x^4 - 280*x^2 + 16, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{18} + 44 x^{16} - 128 x^{14} + 328 x^{12} - 600 x^{10} + 1112 x^{8} - 1472 x^{6} + 996 x^{4} - 280 x^{2} + 16 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(35201711859446669696303104=2^{46}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.94$
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:  $2, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{9} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{14} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{15} - \frac{1}{8} a^{11} + \frac{1}{8} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{17} - \frac{1}{8} a^{9} + \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{112160144} a^{18} + \frac{2339737}{112160144} a^{16} + \frac{34871}{1219132} a^{14} - \frac{1137299}{56080072} a^{12} - \frac{6526807}{56080072} a^{10} + \frac{3188083}{56080072} a^{8} + \frac{766090}{7010009} a^{6} - \frac{1}{2} a^{5} + \frac{1965685}{14020018} a^{4} - \frac{1}{2} a^{3} + \frac{5272005}{14020018} a^{2} - \frac{2227075}{7010009}$, $\frac{1}{112160144} a^{19} + \frac{2339737}{112160144} a^{17} + \frac{34871}{1219132} a^{15} - \frac{1137299}{56080072} a^{13} - \frac{6526807}{56080072} a^{11} + \frac{3188083}{56080072} a^{9} + \frac{766090}{7010009} a^{7} - \frac{1}{2} a^{6} + \frac{1965685}{14020018} a^{5} - \frac{1}{2} a^{4} + \frac{5272005}{14020018} a^{3} - \frac{2227075}{7010009} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 200354.569223 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T230:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1920
The 24 conjugacy class representatives for t20n230
Character table for t20n230 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.1.53824.1, 10.2.1483275763712.1, 10.2.370818940928.1, 10.2.185409470464.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$