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Properties

Label 20.0.48739661569...3529.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{6}\cdot 401^{8}$
Root discriminant $15.29$
Ramified primes $3, 401$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^4:D_5$ (as 20T85)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 0, -2, 3, -6, 11, 10, 15, -19, -15, -19, 15, 10, 11, -6, 3, -2, 0, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 2*x^17 + 3*x^16 - 6*x^15 + 11*x^14 + 10*x^13 + 15*x^12 - 19*x^11 - 15*x^10 - 19*x^9 + 15*x^8 + 10*x^7 + 11*x^6 - 6*x^5 + 3*x^4 - 2*x^3 - x + 1)
 
gp: K = bnfinit(x^20 - x^19 - 2*x^17 + 3*x^16 - 6*x^15 + 11*x^14 + 10*x^13 + 15*x^12 - 19*x^11 - 15*x^10 - 19*x^9 + 15*x^8 + 10*x^7 + 11*x^6 - 6*x^5 + 3*x^4 - 2*x^3 - x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 2 x^{17} + 3 x^{16} - 6 x^{15} + 11 x^{14} + 10 x^{13} + 15 x^{12} - 19 x^{11} - 15 x^{10} - 19 x^{9} + 15 x^{8} + 10 x^{7} + 11 x^{6} - 6 x^{5} + 3 x^{4} - 2 x^{3} - x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(487396615698744004253529=3^{6}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.29$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{15} - \frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{3} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{2}{9} a^{8} - \frac{2}{9} a^{7} - \frac{4}{9} a^{6} + \frac{1}{3} a^{4} + \frac{1}{9} a^{3} + \frac{2}{9} a^{2} + \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{15} - \frac{1}{9} a^{14} - \frac{1}{9} a^{13} + \frac{2}{9} a^{11} + \frac{2}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{2}{9} a^{7} + \frac{4}{9} a^{6} + \frac{4}{9} a^{4} + \frac{1}{9} a^{3} - \frac{1}{9} a^{2} - \frac{1}{3} a + \frac{2}{9}$, $\frac{1}{2241} a^{18} - \frac{14}{747} a^{17} - \frac{22}{2241} a^{16} + \frac{65}{747} a^{15} - \frac{251}{2241} a^{14} - \frac{368}{2241} a^{13} + \frac{161}{2241} a^{12} + \frac{998}{2241} a^{11} - \frac{53}{249} a^{10} + \frac{287}{747} a^{9} + \frac{113}{249} a^{8} + \frac{749}{2241} a^{7} - \frac{337}{2241} a^{6} + \frac{628}{2241} a^{5} - \frac{500}{2241} a^{4} + \frac{65}{747} a^{3} + \frac{476}{2241} a^{2} - \frac{346}{747} a - \frac{995}{2241}$, $\frac{1}{6723} a^{19} + \frac{1}{6723} a^{18} - \frac{85}{6723} a^{17} - \frac{253}{6723} a^{16} - \frac{10}{81} a^{15} + \frac{44}{6723} a^{14} + \frac{340}{2241} a^{13} - \frac{296}{6723} a^{12} - \frac{1138}{6723} a^{11} - \frac{76}{2241} a^{10} - \frac{19}{2241} a^{9} + \frac{1901}{6723} a^{8} - \frac{251}{6723} a^{7} + \frac{86}{249} a^{6} - \frac{1135}{6723} a^{5} - \frac{887}{6723} a^{4} - \frac{103}{6723} a^{3} + \frac{1502}{6723} a^{2} + \frac{1183}{6723} a - \frac{1451}{6723}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6944.31732075 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T85):

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

5.5.160801.1, 10.4.698137963227.1, 10.4.77570884803.1, 10.2.232712654409.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{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}$
401Data not computed