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Properties

Label 20.2.57678015362...0000.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{16}\cdot 5^{10}\cdot 11\cdot 19^{8}\cdot 29^{6}\cdot 811$
Root discriminant $54.71$
Ramified primes $2, 5, 11, 19, 29, 811$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T955

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8921, 0, 32561, 0, 138861, 0, -14709, 0, 55353, 0, 62316, 0, 24159, 0, 4992, 0, 594, 0, 38, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 38*x^18 + 594*x^16 + 4992*x^14 + 24159*x^12 + 62316*x^10 + 55353*x^8 - 14709*x^6 + 138861*x^4 + 32561*x^2 - 8921)
 
gp: K = bnfinit(x^20 + 38*x^18 + 594*x^16 + 4992*x^14 + 24159*x^12 + 62316*x^10 + 55353*x^8 - 14709*x^6 + 138861*x^4 + 32561*x^2 - 8921, 1)
 

Normalized defining polynomial

\( x^{20} + 38 x^{18} + 594 x^{16} + 4992 x^{14} + 24159 x^{12} + 62316 x^{10} + 55353 x^{8} - 14709 x^{6} + 138861 x^{4} + 32561 x^{2} - 8921 \)

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:  \(-57678015362481878140979840000000000=-\,2^{16}\cdot 5^{10}\cdot 11\cdot 19^{8}\cdot 29^{6}\cdot 811\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.71$
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:  $2, 5, 11, 19, 29, 811$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{4} + \frac{1}{9} a^{2} - \frac{2}{9}$, $\frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{2}{9} a$, $\frac{1}{27} a^{6} - \frac{1}{9} a^{2} + \frac{2}{27}$, $\frac{1}{27} a^{7} - \frac{1}{9} a^{3} + \frac{2}{27} a$, $\frac{1}{81} a^{8} - \frac{1}{81} a^{6} - \frac{1}{27} a^{4} + \frac{5}{81} a^{2} - \frac{2}{81}$, $\frac{1}{81} a^{9} - \frac{1}{81} a^{7} - \frac{1}{27} a^{5} + \frac{5}{81} a^{3} - \frac{2}{81} a$, $\frac{1}{243} a^{10} + \frac{1}{243} a^{8} + \frac{4}{243} a^{6} - \frac{1}{243} a^{4} - \frac{19}{243} a^{2} + \frac{14}{243}$, $\frac{1}{243} a^{11} + \frac{1}{243} a^{9} + \frac{4}{243} a^{7} - \frac{1}{243} a^{5} - \frac{19}{243} a^{3} + \frac{14}{243} a$, $\frac{1}{729} a^{12} + \frac{1}{243} a^{8} - \frac{5}{729} a^{6} - \frac{2}{81} a^{4} + \frac{11}{243} a^{2} - \frac{14}{729}$, $\frac{1}{729} a^{13} + \frac{1}{243} a^{9} - \frac{5}{729} a^{7} - \frac{2}{81} a^{5} + \frac{11}{243} a^{3} - \frac{14}{729} a$, $\frac{1}{2187} a^{14} - \frac{1}{2187} a^{12} + \frac{1}{729} a^{10} - \frac{8}{2187} a^{8} - \frac{13}{2187} a^{6} + \frac{17}{729} a^{4} - \frac{47}{2187} a^{2} + \frac{14}{2187}$, $\frac{1}{2187} a^{15} - \frac{1}{2187} a^{13} + \frac{1}{729} a^{11} - \frac{8}{2187} a^{9} - \frac{13}{2187} a^{7} + \frac{17}{729} a^{5} - \frac{47}{2187} a^{3} + \frac{14}{2187} a$, $\frac{1}{6561} a^{16} + \frac{1}{6561} a^{14} + \frac{1}{6561} a^{12} - \frac{2}{6561} a^{10} - \frac{29}{6561} a^{8} + \frac{25}{6561} a^{6} + \frac{55}{6561} a^{4} - \frac{80}{6561} a^{2} + \frac{28}{6561}$, $\frac{1}{6561} a^{17} + \frac{1}{6561} a^{15} + \frac{1}{6561} a^{13} - \frac{2}{6561} a^{11} - \frac{29}{6561} a^{9} + \frac{25}{6561} a^{7} + \frac{55}{6561} a^{5} - \frac{80}{6561} a^{3} + \frac{28}{6561} a$, $\frac{1}{36964674} a^{18} - \frac{1}{13122} a^{17} - \frac{28}{2053593} a^{16} - \frac{1}{13122} a^{15} - \frac{569}{4107186} a^{14} + \frac{4}{6561} a^{13} + \frac{5699}{12321558} a^{12} - \frac{25}{13122} a^{11} + \frac{4087}{4107186} a^{10} - \frac{26}{6561} a^{9} - \frac{8153}{4107186} a^{8} - \frac{97}{13122} a^{7} + \frac{7159}{12321558} a^{6} - \frac{338}{6561} a^{5} - \frac{26365}{2053593} a^{4} - \frac{122}{6561} a^{3} - \frac{651005}{4107186} a^{2} - \frac{5473}{13122} a + \frac{16651151}{36964674}$, $\frac{1}{36964674} a^{19} + \frac{257}{4107186} a^{17} - \frac{1}{13122} a^{16} - \frac{128}{2053593} a^{15} - \frac{1}{13122} a^{14} - \frac{1813}{12321558} a^{13} + \frac{4}{6561} a^{12} - \frac{2495}{2053593} a^{11} - \frac{25}{13122} a^{10} - \frac{8779}{4107186} a^{9} - \frac{26}{6561} a^{8} - \frac{52291}{6160779} a^{7} - \frac{97}{13122} a^{6} + \frac{87880}{2053593} a^{5} - \frac{338}{6561} a^{4} - \frac{253495}{4107186} a^{3} - \frac{122}{6561} a^{2} - \frac{3512867}{18482337} a - \frac{5473}{13122}$

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:  \( 3388982173.69 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T955:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 819200
The 275 conjugacy class representatives for t20n955 are not computed
Character table for t20n955 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.9932496465625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.0.1$x^{8} + x^{2} - 2 x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.8.0.1$x^{8} - x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
811Data not computed