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Properties

Label 20.14.6976189580...0000.1
Degree $20$
Signature $[14, 3]$
Discriminant $-\,2^{44}\cdot 5^{10}\cdot 67^{8}$
Root discriminant $55.23$
Ramified primes $2, 5, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T803

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 124, 0, -3951, 0, 24504, 0, -11822, 0, -28592, 0, 542, 0, 1736, 0, -129, 0, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 12*x^18 - 129*x^16 + 1736*x^14 + 542*x^12 - 28592*x^10 - 11822*x^8 + 24504*x^6 - 3951*x^4 + 124*x^2 - 1)
 
gp: K = bnfinit(x^20 - 12*x^18 - 129*x^16 + 1736*x^14 + 542*x^12 - 28592*x^10 - 11822*x^8 + 24504*x^6 - 3951*x^4 + 124*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 12 x^{18} - 129 x^{16} + 1736 x^{14} + 542 x^{12} - 28592 x^{10} - 11822 x^{8} + 24504 x^{6} - 3951 x^{4} + 124 x^{2} - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-69761895802737851304509440000000000=-\,2^{44}\cdot 5^{10}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.23$
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:  $2, 5, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{3}{8} a^{2} - \frac{3}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} + \frac{3}{8} a^{3} - \frac{3}{8} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{8} - \frac{1}{4} a^{6} - \frac{1}{16} a^{4} + \frac{1}{4} a^{2} + \frac{1}{16}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{9} - \frac{1}{4} a^{7} - \frac{1}{16} a^{5} + \frac{1}{4} a^{3} + \frac{1}{16} a$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{12} - \frac{1}{32} a^{10} - \frac{3}{32} a^{8} - \frac{1}{4} a^{7} + \frac{3}{32} a^{6} - \frac{1}{4} a^{5} + \frac{5}{32} a^{4} + \frac{1}{4} a^{3} - \frac{3}{32} a^{2} + \frac{1}{4} a - \frac{1}{32}$, $\frac{1}{32} a^{15} - \frac{1}{32} a^{13} - \frac{1}{32} a^{11} - \frac{3}{32} a^{9} + \frac{3}{32} a^{7} - \frac{1}{4} a^{6} + \frac{5}{32} a^{5} - \frac{1}{4} a^{4} - \frac{3}{32} a^{3} + \frac{1}{4} a^{2} - \frac{1}{32} a + \frac{1}{4}$, $\frac{1}{32} a^{16} + \frac{1}{16} a^{8} - \frac{1}{2} a^{2} + \frac{13}{32}$, $\frac{1}{32} a^{17} + \frac{1}{16} a^{9} - \frac{1}{2} a^{3} + \frac{13}{32} a$, $\frac{1}{20622823165760} a^{18} + \frac{163387873959}{20622823165760} a^{16} - \frac{675403569}{1031141158288} a^{14} - \frac{141808306411}{5155705791440} a^{12} - \frac{499729775181}{10311411582880} a^{10} - \frac{503957770027}{10311411582880} a^{8} + \frac{743418821621}{5155705791440} a^{6} - \frac{1273639073433}{5155705791440} a^{4} - \frac{1}{2} a^{3} + \frac{4169554076377}{20622823165760} a^{2} - \frac{1}{2} a + \frac{9510278948351}{20622823165760}$, $\frac{1}{20622823165760} a^{19} + \frac{163387873959}{20622823165760} a^{17} - \frac{675403569}{1031141158288} a^{15} - \frac{141808306411}{5155705791440} a^{13} - \frac{499729775181}{10311411582880} a^{11} - \frac{503957770027}{10311411582880} a^{9} + \frac{743418821621}{5155705791440} a^{7} - \frac{1273639073433}{5155705791440} a^{5} + \frac{4169554076377}{20622823165760} a^{3} - \frac{1}{2} a^{2} + \frac{9510278948351}{20622823165760} a - \frac{1}{2}$

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:  $16$
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:  \( 122960983795 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T803:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 126 conjugacy class representatives for t20n803 are not computed
Character table for t20n803 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.5745920.1, 10.10.4126949580800000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$67$67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.12.8.1$x^{12} - 201 x^{9} + 13467 x^{6} - 300763 x^{3} + 161208968$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$