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Properties

Label 20.12.3576211589...0000.4
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 3^{8}\cdot 5^{11}\cdot 239^{8}$
Root discriminant $67.25$
Ramified primes $2, 3, 5, 239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, 0, 300, 0, -9461, 0, -12583, 0, 57220, 0, -40855, 0, 9316, 0, -49, 0, -182, 0, 7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 7*x^18 - 182*x^16 - 49*x^14 + 9316*x^12 - 40855*x^10 + 57220*x^8 - 12583*x^6 - 9461*x^4 + 300*x^2 + 5)
 
gp: K = bnfinit(x^20 + 7*x^18 - 182*x^16 - 49*x^14 + 9316*x^12 - 40855*x^10 + 57220*x^8 - 12583*x^6 - 9461*x^4 + 300*x^2 + 5, 1)
 

Normalized defining polynomial

\( x^{20} + 7 x^{18} - 182 x^{16} - 49 x^{14} + 9316 x^{12} - 40855 x^{10} + 57220 x^{8} - 12583 x^{6} - 9461 x^{4} + 300 x^{2} + 5 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3576211589862019840319539200000000000=2^{20}\cdot 3^{8}\cdot 5^{11}\cdot 239^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.25$
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:  $2, 3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{4}{9} a^{2} - \frac{1}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{4}{9} a^{3} - \frac{1}{9} a$, $\frac{1}{27} a^{14} + \frac{1}{27} a^{12} + \frac{1}{27} a^{10} + \frac{4}{27} a^{8} + \frac{1}{3} a^{6} + \frac{1}{27} a^{4} - \frac{2}{9} a^{2} - \frac{1}{27}$, $\frac{1}{27} a^{15} + \frac{1}{27} a^{13} + \frac{1}{27} a^{11} + \frac{4}{27} a^{9} + \frac{1}{3} a^{7} + \frac{1}{27} a^{5} - \frac{2}{9} a^{3} - \frac{1}{27} a$, $\frac{1}{1215} a^{16} - \frac{19}{1215} a^{14} + \frac{8}{1215} a^{12} - \frac{151}{1215} a^{10} - \frac{25}{243} a^{8} + \frac{199}{1215} a^{6} + \frac{136}{1215} a^{4} - \frac{14}{243} a^{2} + \frac{4}{243}$, $\frac{1}{1215} a^{17} - \frac{19}{1215} a^{15} + \frac{8}{1215} a^{13} - \frac{151}{1215} a^{11} - \frac{25}{243} a^{9} + \frac{199}{1215} a^{7} + \frac{136}{1215} a^{5} - \frac{14}{243} a^{3} + \frac{4}{243} a$, $\frac{1}{105580748274075} a^{18} + \frac{7182735518}{35193582758025} a^{16} + \frac{438970496626}{105580748274075} a^{14} - \frac{1578052287032}{105580748274075} a^{12} + \frac{5544163111199}{35193582758025} a^{10} - \frac{14656068390691}{105580748274075} a^{8} + \frac{11391254051543}{105580748274075} a^{6} - \frac{3195784264223}{11731194252675} a^{4} + \frac{1929253770266}{21116149654815} a^{2} + \frac{3761547370012}{21116149654815}$, $\frac{1}{105580748274075} a^{19} + \frac{7182735518}{35193582758025} a^{17} + \frac{438970496626}{105580748274075} a^{15} - \frac{1578052287032}{105580748274075} a^{13} + \frac{5544163111199}{35193582758025} a^{11} - \frac{14656068390691}{105580748274075} a^{9} + \frac{11391254051543}{105580748274075} a^{7} - \frac{3195784264223}{11731194252675} a^{5} + \frac{1929253770266}{21116149654815} a^{3} + \frac{3761547370012}{21116149654815} a$

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:  $15$
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:  \( 224320069293 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$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$$[\ ]_{2}^{2}$
$5$5.4.3.1$x^{4} - 5$$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}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed