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Properties

Label 20.8.29658361313...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{24}\cdot 5^{13}\cdot 3469^{4}$
Root discriminant $33.39$
Ramified primes $2, 5, 3469$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T755

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![125, 0, 625, 0, -2925, 0, -7950, 0, 225, 0, 3435, 0, 281, 0, -334, 0, -33, 0, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 9*x^18 - 33*x^16 - 334*x^14 + 281*x^12 + 3435*x^10 + 225*x^8 - 7950*x^6 - 2925*x^4 + 625*x^2 + 125)
 
gp: K = bnfinit(x^20 + 9*x^18 - 33*x^16 - 334*x^14 + 281*x^12 + 3435*x^10 + 225*x^8 - 7950*x^6 - 2925*x^4 + 625*x^2 + 125, 1)
 

Normalized defining polynomial

\( x^{20} + 9 x^{18} - 33 x^{16} - 334 x^{14} + 281 x^{12} + 3435 x^{10} + 225 x^{8} - 7950 x^{6} - 2925 x^{4} + 625 x^{2} + 125 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2965836131318190080000000000000=2^{24}\cdot 5^{13}\cdot 3469^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.39$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{10} + \frac{1}{5} a^{8} + \frac{1}{10} a^{6} - \frac{2}{5} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{11} + \frac{1}{5} a^{9} + \frac{1}{10} a^{7} - \frac{2}{5} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{10} a^{14} + \frac{1}{10} a^{10} + \frac{3}{10} a^{8} - \frac{3}{10} a^{6} + \frac{1}{10} a^{4} - \frac{1}{2}$, $\frac{1}{10} a^{15} + \frac{1}{10} a^{11} + \frac{3}{10} a^{9} - \frac{3}{10} a^{7} + \frac{1}{10} a^{5} - \frac{1}{2} a$, $\frac{1}{100} a^{16} + \frac{1}{25} a^{14} + \frac{1}{50} a^{12} - \frac{6}{25} a^{10} + \frac{11}{100} a^{8} + \frac{1}{10} a^{6} - \frac{1}{2} a^{5} + \frac{3}{10} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{100} a^{17} + \frac{1}{25} a^{15} + \frac{1}{50} a^{13} - \frac{6}{25} a^{11} + \frac{11}{100} a^{9} + \frac{1}{10} a^{7} - \frac{1}{2} a^{6} + \frac{3}{10} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{372947718558500} a^{18} + \frac{1538783200949}{372947718558500} a^{16} - \frac{1542877318472}{93236929639625} a^{14} + \frac{871962097943}{186473859279250} a^{12} + \frac{53749400756491}{372947718558500} a^{10} - \frac{1}{2} a^{9} + \frac{15729741428317}{74589543711700} a^{8} - \frac{1}{2} a^{7} - \frac{3410340824087}{18647385927925} a^{6} - \frac{1}{2} a^{5} + \frac{931025453769}{7458954371170} a^{4} - \frac{1532065112623}{14917908742340} a^{2} - \frac{971956673417}{2983581748468}$, $\frac{1}{372947718558500} a^{19} + \frac{1538783200949}{372947718558500} a^{17} - \frac{1542877318472}{93236929639625} a^{15} + \frac{871962097943}{186473859279250} a^{13} + \frac{53749400756491}{372947718558500} a^{11} - \frac{21565030427533}{74589543711700} a^{9} - \frac{3410340824087}{18647385927925} a^{7} - \frac{1399225865908}{3729477185585} a^{5} - \frac{1}{2} a^{4} - \frac{1532065112623}{14917908742340} a^{3} - \frac{1}{2} a^{2} + \frac{519834200817}{2983581748468} a - \frac{1}{2}$

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:  $13$
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:  \( 73363367.7881 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T755:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n755 are not computed
Character table for t20n755 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
3469Data not computed