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Properties

Label 20.2.44322049696...0544.2
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{50}\cdot 89^{8}$
Root discriminant $34.07$
Ramified primes $2, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, -102, 0, -1527, 0, -2882, 0, 837, 0, 902, 0, -724, 0, 42, 0, 34, 0, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^18 + 34*x^16 + 42*x^14 - 724*x^12 + 902*x^10 + 837*x^8 - 2882*x^6 - 1527*x^4 - 102*x^2 - 1)
 
gp: K = bnfinit(x^20 - 8*x^18 + 34*x^16 + 42*x^14 - 724*x^12 + 902*x^10 + 837*x^8 - 2882*x^6 - 1527*x^4 - 102*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{18} + 34 x^{16} + 42 x^{14} - 724 x^{12} + 902 x^{10} + 837 x^{8} - 2882 x^{6} - 1527 x^{4} - 102 x^{2} - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-4432204969617689467720296300544=-\,2^{50}\cdot 89^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.07$
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:  $2, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{12} a^{16} - \frac{1}{3} a^{14} + \frac{1}{12} a^{12} - \frac{1}{2} a^{10} + \frac{1}{4} a^{8} - \frac{1}{3} a^{6} + \frac{1}{6} a^{4} + \frac{1}{6} a^{2} - \frac{5}{12}$, $\frac{1}{12} a^{17} - \frac{1}{3} a^{15} + \frac{1}{12} a^{13} - \frac{1}{2} a^{11} + \frac{1}{4} a^{9} - \frac{1}{3} a^{7} + \frac{1}{6} a^{5} + \frac{1}{6} a^{3} - \frac{5}{12} a$, $\frac{1}{7121263713880632} a^{18} - \frac{51602553474089}{7121263713880632} a^{16} - \frac{1}{2} a^{15} - \frac{2022405865904179}{7121263713880632} a^{14} - \frac{1}{2} a^{13} - \frac{217033327318747}{7121263713880632} a^{12} - \frac{296127161199989}{2373754571293544} a^{10} + \frac{3266643839195729}{7121263713880632} a^{8} - \frac{1}{2} a^{7} + \frac{127193463524871}{1186877285646772} a^{6} - \frac{1}{2} a^{5} - \frac{203080345916521}{593438642823386} a^{4} - \frac{1}{2} a^{3} - \frac{2798144895745519}{7121263713880632} a^{2} - \frac{1}{2} a + \frac{1645655414423717}{7121263713880632}$, $\frac{1}{7121263713880632} a^{19} - \frac{51602553474089}{7121263713880632} a^{17} - \frac{2022405865904179}{7121263713880632} a^{15} - \frac{1}{2} a^{14} - \frac{217033327318747}{7121263713880632} a^{13} - \frac{1}{2} a^{12} - \frac{296127161199989}{2373754571293544} a^{11} + \frac{3266643839195729}{7121263713880632} a^{9} + \frac{127193463524871}{1186877285646772} a^{7} - \frac{1}{2} a^{6} - \frac{203080345916521}{593438642823386} a^{5} - \frac{1}{2} a^{4} - \frac{2798144895745519}{7121263713880632} a^{3} - \frac{1}{2} a^{2} + \frac{1645655414423717}{7121263713880632} a - \frac{1}{2}$

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:  $10$
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:  \( 23928339.409 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n797 are not computed
Character table for t20n797 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.3.31684.1, 10.6.8223751012352.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.8.20.53$x^{8} + 4 x^{5} + 2 x^{4} + 2$$8$$1$$20$$Q_8:C_2$$[2, 3, 3]^{2}$
2.12.30.162$x^{12} - 6 x^{10} - 3 x^{8} + 4 x^{6} - 5 x^{4} + 2 x^{2} - 1$$4$$3$$30$12T134$[2, 2, 2, 3, 7/2, 7/2, 7/2]^{3}$
89Data not computed