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Properties

Label 20.18.1582737989...0000.1
Degree $20$
Signature $[18, 1]$
Discriminant $-\,2^{12}\cdot 5^{10}\cdot 13^{12}\cdot 19^{8}$
Root discriminant $51.28$
Ramified primes $2, 5, 13, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T633

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-256, 0, 5888, 0, -21088, 0, 27140, 0, -13305, 0, -1163, 0, 4019, 0, -1763, 0, 350, 0, -32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 32*x^18 + 350*x^16 - 1763*x^14 + 4019*x^12 - 1163*x^10 - 13305*x^8 + 27140*x^6 - 21088*x^4 + 5888*x^2 - 256)
 
gp: K = bnfinit(x^20 - 32*x^18 + 350*x^16 - 1763*x^14 + 4019*x^12 - 1163*x^10 - 13305*x^8 + 27140*x^6 - 21088*x^4 + 5888*x^2 - 256, 1)
 

Normalized defining polynomial

\( x^{20} - 32 x^{18} + 350 x^{16} - 1763 x^{14} + 4019 x^{12} - 1163 x^{10} - 13305 x^{8} + 27140 x^{6} - 21088 x^{4} + 5888 x^{2} - 256 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 1]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-15827379896489610792988840000000000=-\,2^{12}\cdot 5^{10}\cdot 13^{12}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.28$
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:  $2, 5, 13, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{10} + \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{4} a^{11} - \frac{3}{8} a^{9} + \frac{3}{8} a^{7} - \frac{3}{8} a^{5} - \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{16} - \frac{1}{16} a^{15} - \frac{1}{8} a^{14} - \frac{1}{4} a^{13} - \frac{1}{16} a^{12} + \frac{1}{8} a^{11} + \frac{5}{32} a^{10} - \frac{5}{16} a^{9} - \frac{1}{32} a^{8} - \frac{7}{16} a^{7} + \frac{9}{32} a^{6} - \frac{1}{16} a^{5} - \frac{13}{32} a^{4} + \frac{5}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{64} a^{17} + \frac{7}{32} a^{13} - \frac{1}{2} a^{12} + \frac{29}{64} a^{11} - \frac{13}{64} a^{9} + \frac{5}{64} a^{7} - \frac{1}{2} a^{6} - \frac{9}{64} a^{5} - \frac{1}{2} a^{4} + \frac{5}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{102815549824} a^{18} - \frac{142227093}{12851943728} a^{16} - \frac{4505579257}{51407774912} a^{14} - \frac{1}{4} a^{13} + \frac{2295508717}{102815549824} a^{12} + \frac{33310633323}{102815549824} a^{10} - \frac{1}{2} a^{9} + \frac{31753964109}{102815549824} a^{8} - \frac{1}{4} a^{7} + \frac{61525281}{1301462656} a^{6} + \frac{1}{4} a^{5} - \frac{3627255937}{25703887456} a^{4} - \frac{1}{4} a^{3} - \frac{1408984079}{6425971864} a^{2} + \frac{1}{4} a + \frac{266181062}{803246483}$, $\frac{1}{205631099648} a^{19} - \frac{142227093}{25703887456} a^{17} - \frac{4505579257}{102815549824} a^{15} - \frac{1}{8} a^{14} + \frac{2295508717}{205631099648} a^{13} - \frac{1}{2} a^{12} - \frac{69504916501}{205631099648} a^{11} + \frac{1}{4} a^{10} - \frac{71061585715}{205631099648} a^{9} + \frac{3}{8} a^{8} - \frac{1239937375}{2602925312} a^{7} + \frac{1}{8} a^{6} - \frac{3627255937}{51407774912} a^{5} - \frac{1}{8} a^{4} - \frac{1408984079}{12851943728} a^{3} - \frac{3}{8} a^{2} + \frac{133090531}{803246483} a$

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:  $18$
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:  \( 45959571489.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T633:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40960
The 124 conjugacy class representatives for t20n633 are not computed
Character table for t20n633 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.19827925.1, 10.10.1965733049028125.1

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 $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$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$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$