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Properties

Label 20.20.6853400654...3904.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 3^{10}\cdot 71^{8}\cdot 73\cdot 2293$
Root discriminant $49.18$
Ramified primes $2, 3, 71, 73, 2293$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T647

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![167389, 0, -773118, 0, 1437469, 0, -1411160, 0, 815802, 0, -294020, 0, 67778, 0, -9976, 0, 905, 0, -46, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 46*x^18 + 905*x^16 - 9976*x^14 + 67778*x^12 - 294020*x^10 + 815802*x^8 - 1411160*x^6 + 1437469*x^4 - 773118*x^2 + 167389)
 
gp: K = bnfinit(x^20 - 46*x^18 + 905*x^16 - 9976*x^14 + 67778*x^12 - 294020*x^10 + 815802*x^8 - 1411160*x^6 + 1437469*x^4 - 773118*x^2 + 167389, 1)
 

Normalized defining polynomial

\( x^{20} - 46 x^{18} + 905 x^{16} - 9976 x^{14} + 67778 x^{12} - 294020 x^{10} + 815802 x^{8} - 1411160 x^{6} + 1437469 x^{4} - 773118 x^{2} + 167389 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6853400654600934079309919922683904=2^{30}\cdot 3^{10}\cdot 71^{8}\cdot 73\cdot 2293\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.18$
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:  $2, 3, 71, 73, 2293$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3} + \frac{1}{8} a$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{4} + \frac{1}{16}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{5} + \frac{1}{16} a$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{8} - \frac{1}{16} a^{6} + \frac{1}{16} a^{4} + \frac{1}{32} a^{2} - \frac{1}{32}$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{1}{64} a^{9} + \frac{1}{64} a^{8} - \frac{1}{32} a^{7} + \frac{1}{32} a^{6} - \frac{3}{32} a^{5} + \frac{3}{32} a^{4} + \frac{1}{64} a^{3} - \frac{1}{64} a^{2} - \frac{25}{64} a + \frac{25}{64}$, $\frac{1}{64} a^{12} + \frac{1}{64} a^{8} - \frac{5}{64} a^{4} + \frac{3}{64}$, $\frac{1}{128} a^{13} - \frac{1}{128} a^{12} - \frac{3}{128} a^{9} + \frac{3}{128} a^{8} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{5}{128} a^{5} - \frac{11}{128} a^{4} - \frac{3}{16} a^{3} - \frac{3}{16} a^{2} - \frac{25}{128} a - \frac{23}{128}$, $\frac{1}{128} a^{14} - \frac{1}{128} a^{12} + \frac{1}{128} a^{10} - \frac{1}{128} a^{8} - \frac{5}{128} a^{6} + \frac{5}{128} a^{4} + \frac{3}{128} a^{2} - \frac{3}{128}$, $\frac{1}{256} a^{15} - \frac{1}{256} a^{14} - \frac{1}{256} a^{13} + \frac{1}{256} a^{12} + \frac{1}{256} a^{11} - \frac{1}{256} a^{10} + \frac{7}{256} a^{9} - \frac{7}{256} a^{8} - \frac{5}{256} a^{7} + \frac{5}{256} a^{6} - \frac{11}{256} a^{5} + \frac{11}{256} a^{4} + \frac{3}{256} a^{3} - \frac{3}{256} a^{2} - \frac{123}{256} a + \frac{123}{256}$, $\frac{1}{256} a^{16} - \frac{3}{128} a^{8} + \frac{1}{32} a^{4} - \frac{3}{256}$, $\frac{1}{512} a^{17} - \frac{1}{512} a^{16} + \frac{5}{256} a^{9} - \frac{5}{256} a^{8} + \frac{5}{64} a^{5} - \frac{5}{64} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{77}{512} a + \frac{179}{512}$, $\frac{1}{512} a^{18} - \frac{1}{512} a^{16} - \frac{3}{256} a^{10} + \frac{3}{256} a^{8} + \frac{1}{64} a^{6} - \frac{1}{64} a^{4} - \frac{3}{512} a^{2} + \frac{3}{512}$, $\frac{1}{1024} a^{19} - \frac{1}{1024} a^{18} - \frac{1}{1024} a^{17} + \frac{1}{1024} a^{16} - \frac{3}{512} a^{11} + \frac{3}{512} a^{10} + \frac{3}{512} a^{9} - \frac{3}{512} a^{8} - \frac{7}{128} a^{7} + \frac{7}{128} a^{6} + \frac{7}{128} a^{5} - \frac{7}{128} a^{4} - \frac{195}{1024} a^{3} + \frac{195}{1024} a^{2} - \frac{317}{1024} a + \frac{317}{1024}$

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:  $19$
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:  \( 31004551209.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T647:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 51200
The 152 conjugacy class representatives for t20n647 are not computed
Character table for t20n647 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 10.10.6323239406592.1

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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
3Data not computed
$71$71.5.4.4$x^{5} + 568$$5$$1$$4$$C_5$$[\ ]_{5}$
71.5.4.4$x^{5} + 568$$5$$1$$4$$C_5$$[\ ]_{5}$
71.10.0.1$x^{10} - x + 22$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
73.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
2293Data not computed