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Properties

Label 20.0.28937457747...3989.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{16}\cdot 229^{5}$
Root discriminant $26.49$
Ramified primes $11, 229$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5\times S_4$ (as 20T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -3, 4, 156, 67, -180, -27, 445, -149, -157, 60, 184, -36, -31, 0, 25, -4, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^17 + 25*x^16 - 31*x^14 - 36*x^13 + 184*x^12 + 60*x^11 - 157*x^10 - 149*x^9 + 445*x^8 - 27*x^7 - 180*x^6 + 67*x^5 + 156*x^4 + 4*x^3 - 3*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^17 + 25*x^16 - 31*x^14 - 36*x^13 + 184*x^12 + 60*x^11 - 157*x^10 - 149*x^9 + 445*x^8 - 27*x^7 - 180*x^6 + 67*x^5 + 156*x^4 + 4*x^3 - 3*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{17} + 25 x^{16} - 31 x^{14} - 36 x^{13} + 184 x^{12} + 60 x^{11} - 157 x^{10} - 149 x^{9} + 445 x^{8} - 27 x^{7} - 180 x^{6} + 67 x^{5} + 156 x^{4} + 4 x^{3} - 3 x^{2} - 3 x + 1 \)

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:  \(28937457747213411097802363989=11^{16}\cdot 229^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.49$
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:  $11, 229$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{704744272659015175733983} a^{19} + \frac{142243865261890776997366}{704744272659015175733983} a^{18} - \frac{42883739625596967685796}{704744272659015175733983} a^{17} + \frac{90419823616967930338258}{704744272659015175733983} a^{16} - \frac{340189657335595815325452}{704744272659015175733983} a^{15} + \frac{89087419552096435771409}{704744272659015175733983} a^{14} - \frac{311018256938396316274890}{704744272659015175733983} a^{13} + \frac{167400922710959000513785}{704744272659015175733983} a^{12} + \frac{46812060642796632046663}{704744272659015175733983} a^{11} - \frac{151587072637686772306168}{704744272659015175733983} a^{10} + \frac{158931728372655276745789}{704744272659015175733983} a^{9} - \frac{217067947154451448431258}{704744272659015175733983} a^{8} - \frac{276209649862013697605058}{704744272659015175733983} a^{7} - \frac{22949777529630424342164}{704744272659015175733983} a^{6} - \frac{113046323806711838991523}{704744272659015175733983} a^{5} + \frac{200686915707621101913755}{704744272659015175733983} a^{4} + \frac{114802820051412838549232}{704744272659015175733983} a^{3} - \frac{191005206406748436001988}{704744272659015175733983} a^{2} - \frac{192165076886442838000504}{704744272659015175733983} a - \frac{151474995571246581498479}{704744272659015175733983}$

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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 957497.704734 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times S_4$ (as 20T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 25 conjugacy class representatives for $C_5\times S_4$
Character table for $C_5\times S_4$ is not computed

Intermediate fields

4.0.229.1, \(\Q(\zeta_{11})^+\)

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

Sibling fields

Degree 30 siblings: data not computed
Degree 40 sibling: 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 $20$ $15{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ $20$ R $20$ $15{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
11Data not computed
229Data not computed