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Properties

Label 20.8.26774124053...0000.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 5^{13}\cdot 3803^{4}$
Root discriminant $29.61$
Ramified primes $2, 5, 3803$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T802

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![125, 0, -125, 0, -1525, 0, 3850, 0, -3685, 0, 1615, 0, -161, 0, -157, 0, 75, 0, -14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 14*x^18 + 75*x^16 - 157*x^14 - 161*x^12 + 1615*x^10 - 3685*x^8 + 3850*x^6 - 1525*x^4 - 125*x^2 + 125)
 
gp: K = bnfinit(x^20 - 14*x^18 + 75*x^16 - 157*x^14 - 161*x^12 + 1615*x^10 - 3685*x^8 + 3850*x^6 - 1525*x^4 - 125*x^2 + 125, 1)
 

Normalized defining polynomial

\( x^{20} - 14 x^{18} + 75 x^{16} - 157 x^{14} - 161 x^{12} + 1615 x^{10} - 3685 x^{8} + 3850 x^{6} - 1525 x^{4} - 125 x^{2} + 125 \)

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:  \(267741240538215680000000000000=2^{20}\cdot 5^{13}\cdot 3803^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.61$
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:  $2, 5, 3803$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{10} - \frac{2}{5} a^{6} - \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{11} - \frac{2}{5} a^{7} - \frac{1}{5} a^{5}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{10} - \frac{2}{5} a^{8} + \frac{1}{5} a^{6} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{11} - \frac{2}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{5}$, $\frac{1}{175} a^{16} + \frac{1}{175} a^{14} + \frac{3}{35} a^{12} - \frac{1}{25} a^{10} + \frac{9}{175} a^{8} + \frac{3}{35} a^{4} + \frac{2}{7} a^{2} + \frac{1}{7}$, $\frac{1}{175} a^{17} + \frac{1}{175} a^{15} + \frac{3}{35} a^{13} - \frac{1}{25} a^{11} + \frac{9}{175} a^{9} + \frac{3}{35} a^{5} + \frac{2}{7} a^{3} + \frac{1}{7} a$, $\frac{1}{165751075} a^{18} - \frac{68321}{165751075} a^{16} - \frac{13060587}{165751075} a^{14} - \frac{14752922}{165751075} a^{12} + \frac{41814558}{165751075} a^{10} - \frac{31007463}{165751075} a^{8} - \frac{11582274}{33150215} a^{6} + \frac{2775974}{6630043} a^{4} + \frac{2668019}{6630043} a^{2} - \frac{2487599}{6630043}$, $\frac{1}{165751075} a^{19} - \frac{68321}{165751075} a^{17} - \frac{13060587}{165751075} a^{15} - \frac{14752922}{165751075} a^{13} + \frac{41814558}{165751075} a^{11} - \frac{31007463}{165751075} a^{9} - \frac{11582274}{33150215} a^{7} + \frac{2775974}{6630043} a^{5} + \frac{2668019}{6630043} a^{3} - \frac{2487599}{6630043} a$

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:  $13$
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:  \( 13278904.0758 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T802:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n802 are not computed
Character table for t20n802 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.3.19015.1, 10.6.45196278125.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
3803Data not computed