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Properties

Label 20.8.61571735716...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{16}\cdot 3^{14}\cdot 5^{12}\cdot 691^{4}\cdot 3529$
Root discriminant $54.89$
Ramified primes $2, 3, 5, 691, 3529$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1037

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31761, 0, -32598, 0, -5427, 0, 11688, 0, 786, 0, -2388, 0, 50, 0, 264, 0, -35, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^18 - 35*x^16 + 264*x^14 + 50*x^12 - 2388*x^10 + 786*x^8 + 11688*x^6 - 5427*x^4 - 32598*x^2 + 31761)
 
gp: K = bnfinit(x^20 - 6*x^18 - 35*x^16 + 264*x^14 + 50*x^12 - 2388*x^10 + 786*x^8 + 11688*x^6 - 5427*x^4 - 32598*x^2 + 31761, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{18} - 35 x^{16} + 264 x^{14} + 50 x^{12} - 2388 x^{10} + 786 x^{8} + 11688 x^{6} - 5427 x^{4} - 32598 x^{2} + 31761 \)

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:  \(61571735716086245423376000000000000=2^{16}\cdot 3^{14}\cdot 5^{12}\cdot 691^{4}\cdot 3529\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.89$
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:  $2, 3, 5, 691, 3529$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3} + \frac{1}{8} a$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{4} + \frac{1}{16}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{5} + \frac{1}{16} a$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{8} - \frac{1}{16} a^{6} + \frac{1}{16} a^{4} + \frac{1}{32} a^{2} - \frac{1}{32}$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{9} - \frac{1}{16} a^{7} + \frac{1}{16} a^{5} + \frac{1}{32} a^{3} - \frac{1}{32} a$, $\frac{1}{64} a^{12} + \frac{1}{64} a^{8} - \frac{5}{64} a^{4} + \frac{3}{64}$, $\frac{1}{64} a^{13} + \frac{1}{64} a^{9} - \frac{5}{64} a^{5} + \frac{3}{64} a$, $\frac{1}{384} a^{14} - \frac{1}{128} a^{12} + \frac{1}{384} a^{10} - \frac{1}{128} a^{8} + \frac{11}{384} a^{6} - \frac{11}{128} a^{4} + \frac{17}{128} a^{2} - \frac{51}{128}$, $\frac{1}{384} a^{15} - \frac{1}{128} a^{13} + \frac{1}{384} a^{11} - \frac{1}{128} a^{9} + \frac{11}{384} a^{7} - \frac{11}{128} a^{5} + \frac{17}{128} a^{3} - \frac{51}{128} a$, $\frac{1}{768} a^{16} + \frac{1}{192} a^{12} + \frac{7}{384} a^{8} + \frac{7}{64} a^{4} + \frac{51}{256}$, $\frac{1}{1536} a^{17} - \frac{1}{1536} a^{16} - \frac{1}{192} a^{13} + \frac{1}{192} a^{12} - \frac{1}{64} a^{11} - \frac{1}{64} a^{10} + \frac{13}{768} a^{9} + \frac{11}{768} a^{8} - \frac{1}{32} a^{7} - \frac{1}{32} a^{6} + \frac{1}{16} a^{4} + \frac{3}{64} a^{3} + \frac{3}{64} a^{2} + \frac{79}{512} a - \frac{127}{512}$, $\frac{1}{408576} a^{18} - \frac{5}{19456} a^{16} - \frac{5}{7296} a^{14} + \frac{51}{8512} a^{12} - \frac{661}{68096} a^{10} + \frac{1201}{68096} a^{8} - \frac{295}{6384} a^{6} - \frac{1433}{17024} a^{4} + \frac{26567}{136192} a^{2} + \frac{40281}{136192}$, $\frac{1}{817152} a^{19} - \frac{1}{817152} a^{18} - \frac{5}{38912} a^{17} + \frac{5}{38912} a^{16} - \frac{5}{14592} a^{15} + \frac{5}{14592} a^{14} + \frac{51}{17024} a^{13} - \frac{51}{17024} a^{12} - \frac{661}{136192} a^{11} + \frac{661}{136192} a^{10} + \frac{1201}{136192} a^{9} - \frac{1201}{136192} a^{8} - \frac{295}{12768} a^{7} + \frac{295}{12768} a^{6} + \frac{2823}{34048} a^{5} - \frac{2823}{34048} a^{4} - \frac{41529}{272384} a^{3} + \frac{41529}{272384} a^{2} - \frac{61863}{272384} a + \frac{61863}{272384}$

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:  \( 6142181976.58 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1037:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 384 conjugacy class representatives for t20n1037 are not computed
Character table for t20n1037 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.5438807015625.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ $16{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
3.12.14.12$x^{12} + 9 x^{11} + 9 x^{10} + 12 x^{8} + 12 x^{6} + 9 x^{5} + 9 x^{4} - 9 x^{2} - 9$$6$$2$$14$12T39$[3/2, 3/2]_{2}^{4}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
691Data not computed
3529Data not computed