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Properties

Label 20.8.43898071599...2321.1
Degree $20$
Signature $[8, 6]$
Discriminant $3^{4}\cdot 71^{2}\cdot 401^{10}$
Root discriminant $38.21$
Ramified primes $3, 71, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T347

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -216, 124, 936, -1155, -2190, 807, 1430, 1265, 100, -2574, 454, 1377, -814, 7, 202, -113, 16, 11, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 11*x^18 + 16*x^17 - 113*x^16 + 202*x^15 + 7*x^14 - 814*x^13 + 1377*x^12 + 454*x^11 - 2574*x^10 + 100*x^9 + 1265*x^8 + 1430*x^7 + 807*x^6 - 2190*x^5 - 1155*x^4 + 936*x^3 + 124*x^2 - 216*x + 81)
 
gp: K = bnfinit(x^20 - 6*x^19 + 11*x^18 + 16*x^17 - 113*x^16 + 202*x^15 + 7*x^14 - 814*x^13 + 1377*x^12 + 454*x^11 - 2574*x^10 + 100*x^9 + 1265*x^8 + 1430*x^7 + 807*x^6 - 2190*x^5 - 1155*x^4 + 936*x^3 + 124*x^2 - 216*x + 81, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 11 x^{18} + 16 x^{17} - 113 x^{16} + 202 x^{15} + 7 x^{14} - 814 x^{13} + 1377 x^{12} + 454 x^{11} - 2574 x^{10} + 100 x^{9} + 1265 x^{8} + 1430 x^{7} + 807 x^{6} - 2190 x^{5} - 1155 x^{4} + 936 x^{3} + 124 x^{2} - 216 x + 81 \)

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:  \(43898071599567621806622824892321=3^{4}\cdot 71^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.21$
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:  $3, 71, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{13} + \frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{3} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{3} a^{9} - \frac{1}{2} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{6} a^{2} + \frac{1}{3} a$, $\frac{1}{18} a^{16} - \frac{1}{18} a^{14} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{4}{9} a^{8} - \frac{1}{18} a^{6} + \frac{1}{6} a^{5} + \frac{4}{9} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{7}{18} a$, $\frac{1}{18} a^{17} - \frac{1}{18} a^{15} - \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{4}{9} a^{9} - \frac{1}{18} a^{7} + \frac{1}{6} a^{6} + \frac{4}{9} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{7}{18} a^{2}$, $\frac{1}{162} a^{18} - \frac{1}{81} a^{17} - \frac{1}{162} a^{16} - \frac{7}{162} a^{15} - \frac{1}{81} a^{14} - \frac{7}{54} a^{13} + \frac{13}{54} a^{12} - \frac{17}{81} a^{11} - \frac{11}{81} a^{10} - \frac{19}{81} a^{9} - \frac{19}{162} a^{8} + \frac{34}{81} a^{7} + \frac{47}{162} a^{6} - \frac{31}{162} a^{5} - \frac{5}{18} a^{4} - \frac{16}{81} a^{3} - \frac{77}{162} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{2129052556303778192958} a^{19} + \frac{2763722433226296391}{1064526278151889096479} a^{18} - \frac{5779888323872414197}{354842092717296365493} a^{17} - \frac{9602353359651639227}{2129052556303778192958} a^{16} - \frac{2926076464890764231}{709684185434592730986} a^{15} - \frac{41217446781582200315}{2129052556303778192958} a^{14} - \frac{8367249465354194173}{78853798381621414554} a^{13} + \frac{196388372226287272039}{1064526278151889096479} a^{12} - \frac{80217453870908788985}{2129052556303778192958} a^{11} - \frac{84631694594497657855}{709684185434592730986} a^{10} + \frac{48779647196721255365}{354842092717296365493} a^{9} + \frac{578883267021440979151}{2129052556303778192958} a^{8} + \frac{969888161271447922519}{2129052556303778192958} a^{7} - \frac{316942906257069506671}{1064526278151889096479} a^{6} + \frac{317364954806747884702}{1064526278151889096479} a^{5} + \frac{623901464087908790497}{2129052556303778192958} a^{4} - \frac{46120216886179274}{1064526278151889096479} a^{3} + \frac{395124497183337936656}{1064526278151889096479} a^{2} - \frac{12048264594627179062}{39426899190810707277} a + \frac{2344001677087771352}{13142299730270235759}$

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:  \( 246062176.867 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T347:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n347 are not computed
Character table for t20n347 is not computed

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$71$71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.2.2$x^{4} - 71 x^{2} + 55451$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
71.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed