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Properties

Label 20.6.31749116102...5523.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,19^{8}\cdot 83^{9}$
Root discriminant $23.72$
Ramified primes $19, 83$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times S_5$ (as 20T65)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-64, -128, -228, -339, -411, -492, -504, -430, -237, -268, -202, -32, 44, -89, -67, 59, 61, -6, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 15*x^18 - 6*x^17 + 61*x^16 + 59*x^15 - 67*x^14 - 89*x^13 + 44*x^12 - 32*x^11 - 202*x^10 - 268*x^9 - 237*x^8 - 430*x^7 - 504*x^6 - 492*x^5 - 411*x^4 - 339*x^3 - 228*x^2 - 128*x - 64)
 
gp: K = bnfinit(x^20 - 15*x^18 - 6*x^17 + 61*x^16 + 59*x^15 - 67*x^14 - 89*x^13 + 44*x^12 - 32*x^11 - 202*x^10 - 268*x^9 - 237*x^8 - 430*x^7 - 504*x^6 - 492*x^5 - 411*x^4 - 339*x^3 - 228*x^2 - 128*x - 64, 1)
 

Normalized defining polynomial

\( x^{20} - 15 x^{18} - 6 x^{17} + 61 x^{16} + 59 x^{15} - 67 x^{14} - 89 x^{13} + 44 x^{12} - 32 x^{11} - 202 x^{10} - 268 x^{9} - 237 x^{8} - 430 x^{7} - 504 x^{6} - 492 x^{5} - 411 x^{4} - 339 x^{3} - 228 x^{2} - 128 x - 64 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-3174911610236904755365045523=-\,19^{8}\cdot 83^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.72$
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:  $19, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} + \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{68} a^{18} + \frac{3}{34} a^{17} + \frac{3}{68} a^{16} + \frac{3}{34} a^{15} - \frac{2}{17} a^{14} + \frac{5}{68} a^{13} + \frac{5}{68} a^{12} + \frac{1}{17} a^{11} - \frac{5}{68} a^{10} - \frac{15}{68} a^{9} - \frac{15}{68} a^{8} + \frac{7}{34} a^{7} + \frac{15}{68} a^{6} - \frac{7}{34} a^{5} + \frac{9}{68} a^{4} + \frac{9}{34} a^{3} + \frac{11}{68} a^{2} - \frac{1}{34} a + \frac{1}{17}$, $\frac{1}{47558875996500902576} a^{19} + \frac{12432016287956619}{11889718999125225644} a^{18} - \frac{2459522138326169595}{47558875996500902576} a^{17} + \frac{120990676457829437}{1398790470485320664} a^{16} - \frac{3891656242294097659}{47558875996500902576} a^{15} + \frac{2072423136613490971}{47558875996500902576} a^{14} - \frac{326251455529093191}{47558875996500902576} a^{13} - \frac{2162762538620232845}{47558875996500902576} a^{12} + \frac{1311324121260659857}{2972429749781306411} a^{11} - \frac{5181140124458857753}{11889718999125225644} a^{10} - \frac{9730951647522277369}{23779437998250451288} a^{9} + \frac{3632816081722701289}{11889718999125225644} a^{8} + \frac{19988656919613358895}{47558875996500902576} a^{7} + \frac{6429622844327374141}{23779437998250451288} a^{6} - \frac{133096554425279544}{2972429749781306411} a^{5} - \frac{78976643160432501}{174848808810665083} a^{4} - \frac{1972198596725574631}{47558875996500902576} a^{3} + \frac{3869374147839693529}{47558875996500902576} a^{2} - \frac{833907760591435617}{5944859499562612822} a + \frac{315883934452830297}{2972429749781306411}$

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:  $12$
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:  \( 1253288.90185 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_5$ (as 20T65):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 240
The 14 conjugacy class representatives for $C_2\times S_5$
Character table for $C_2\times S_5$

Intermediate fields

10.4.74515853627.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 12 siblings: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 30 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$19$19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
83Data not computed