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Properties

Label 20.20.4239038862...8576.2
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 3^{10}\cdot 401^{8}$
Root discriminant $53.87$
Ramified primes $2, 3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:D_5$ (as 20T73)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 0, -3672, 0, 45846, 0, -185154, 0, 293479, 0, -209152, 0, 70579, 0, -12352, 0, 1151, 0, -54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 54*x^18 + 1151*x^16 - 12352*x^14 + 70579*x^12 - 209152*x^10 + 293479*x^8 - 185154*x^6 + 45846*x^4 - 3672*x^2 + 81)
 
gp: K = bnfinit(x^20 - 54*x^18 + 1151*x^16 - 12352*x^14 + 70579*x^12 - 209152*x^10 + 293479*x^8 - 185154*x^6 + 45846*x^4 - 3672*x^2 + 81, 1)
 

Normalized defining polynomial

\( x^{20} - 54 x^{18} + 1151 x^{16} - 12352 x^{14} + 70579 x^{12} - 209152 x^{10} + 293479 x^{8} - 185154 x^{6} + 45846 x^{4} - 3672 x^{2} + 81 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(42390388623295510152536086938648576=2^{30}\cdot 3^{10}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.87$
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:  $2, 3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{10} - \frac{1}{6} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{14} + \frac{1}{6} a^{11} + \frac{1}{12} a^{10} - \frac{1}{3} a^{9} - \frac{5}{12} a^{8} - \frac{5}{12} a^{7} + \frac{1}{6} a^{6} - \frac{1}{12} a^{5} - \frac{1}{6} a^{4} - \frac{5}{12} a^{3} - \frac{1}{12} a^{2} + \frac{1}{4}$, $\frac{1}{36} a^{16} - \frac{1}{12} a^{14} + \frac{2}{9} a^{12} - \frac{1}{4} a^{11} - \frac{1}{36} a^{10} - \frac{1}{4} a^{9} + \frac{1}{9} a^{8} + \frac{1}{4} a^{7} + \frac{5}{36} a^{6} + \frac{1}{4} a^{5} - \frac{17}{36} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{36} a^{17} - \frac{1}{12} a^{14} + \frac{2}{9} a^{13} - \frac{1}{4} a^{12} + \frac{5}{36} a^{11} - \frac{1}{6} a^{10} - \frac{2}{9} a^{9} - \frac{1}{6} a^{8} - \frac{5}{18} a^{7} + \frac{5}{12} a^{6} + \frac{4}{9} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{293105306259324684} a^{18} - \frac{193115599560893}{97701768753108228} a^{16} - \frac{1348666474156720}{73276326564831171} a^{14} - \frac{1}{4} a^{13} + \frac{14036290246405109}{293105306259324684} a^{12} - \frac{1}{4} a^{11} + \frac{3510659468959411}{73276326564831171} a^{10} + \frac{1}{4} a^{9} - \frac{97013975131563883}{293105306259324684} a^{8} + \frac{1}{4} a^{7} + \frac{106138313999933611}{293105306259324684} a^{6} - \frac{1}{2} a^{5} - \frac{47567430600725917}{97701768753108228} a^{4} - \frac{1}{4} a^{3} - \frac{4586634384997223}{32567256251036076} a^{2} + \frac{2068112514086343}{5427876041839346}$, $\frac{1}{293105306259324684} a^{19} - \frac{193115599560893}{97701768753108228} a^{17} - \frac{1348666474156720}{73276326564831171} a^{15} - \frac{1}{12} a^{14} + \frac{14036290246405109}{293105306259324684} a^{13} - \frac{1}{4} a^{12} + \frac{3510659468959411}{73276326564831171} a^{11} + \frac{1}{12} a^{10} - \frac{97013975131563883}{293105306259324684} a^{9} + \frac{1}{12} a^{8} + \frac{106138313999933611}{293105306259324684} a^{7} + \frac{1}{6} a^{6} - \frac{47567430600725917}{97701768753108228} a^{5} + \frac{1}{12} a^{4} - \frac{4586634384997223}{32567256251036076} a^{3} + \frac{1}{6} a^{2} + \frac{2068112514086343}{5427876041839346} a - \frac{1}{2}$

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:  \( 152047329101 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T73):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

\(\Q(\sqrt{3}) \), 5.5.160801.1, 10.10.714893274344448.1, 10.10.6434039469100032.1, 10.10.238297758114816.1

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 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
401Data not computed