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Properties

Label 20.20.1186334452...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{28}\cdot 5^{15}\cdot 3469^{4}$
Root discriminant $45.05$
Ramified primes $2, 5, 3469$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_5^2:C_4$ (as 20T93)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -288, 512, 5632, -15984, -22864, 104592, -52928, -114392, 103920, 46636, -64384, -6340, 18976, -704, -2868, 296, 212, -30, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 30*x^18 + 212*x^17 + 296*x^16 - 2868*x^15 - 704*x^14 + 18976*x^13 - 6340*x^12 - 64384*x^11 + 46636*x^10 + 103920*x^9 - 114392*x^8 - 52928*x^7 + 104592*x^6 - 22864*x^5 - 15984*x^4 + 5632*x^3 + 512*x^2 - 288*x + 16)
 
gp: K = bnfinit(x^20 - 6*x^19 - 30*x^18 + 212*x^17 + 296*x^16 - 2868*x^15 - 704*x^14 + 18976*x^13 - 6340*x^12 - 64384*x^11 + 46636*x^10 + 103920*x^9 - 114392*x^8 - 52928*x^7 + 104592*x^6 - 22864*x^5 - 15984*x^4 + 5632*x^3 + 512*x^2 - 288*x + 16, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 30 x^{18} + 212 x^{17} + 296 x^{16} - 2868 x^{15} - 704 x^{14} + 18976 x^{13} - 6340 x^{12} - 64384 x^{11} + 46636 x^{10} + 103920 x^{9} - 114392 x^{8} - 52928 x^{7} + 104592 x^{6} - 22864 x^{5} - 15984 x^{4} + 5632 x^{3} + 512 x^{2} - 288 x + 16 \)

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:  \(1186334452527276032000000000000000=2^{28}\cdot 5^{15}\cdot 3469^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.05$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{4} a^{13}$, $\frac{1}{8} a^{14} - \frac{1}{4} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{8} a^{15}$, $\frac{1}{8} a^{16}$, $\frac{1}{8} a^{17}$, $\frac{1}{8} a^{18}$, $\frac{1}{75780985902749966702704} a^{19} - \frac{129983521730389824265}{9472623237843745837838} a^{18} - \frac{2228087776990244978639}{37890492951374983351352} a^{17} - \frac{521444099095929029953}{37890492951374983351352} a^{16} + \frac{2203239628375323004377}{37890492951374983351352} a^{15} - \frac{2217317654895814009895}{37890492951374983351352} a^{14} + \frac{734469549176742528131}{9472623237843745837838} a^{13} + \frac{895213178468983671345}{9472623237843745837838} a^{12} + \frac{159480356263961478019}{9472623237843745837838} a^{11} + \frac{17381812393426579905}{18945246475687491675676} a^{10} - \frac{3705599917144246301987}{18945246475687491675676} a^{9} + \frac{1110545240445079344687}{9472623237843745837838} a^{8} - \frac{415092564257325668292}{4736311618921872918919} a^{7} - \frac{860300125969408197042}{4736311618921872918919} a^{6} + \frac{2146784777258373567233}{9472623237843745837838} a^{5} + \frac{1188405382116774706858}{4736311618921872918919} a^{4} + \frac{1370515938264818445161}{4736311618921872918919} a^{3} + \frac{184430255731220281854}{4736311618921872918919} a^{2} + \frac{1909186873580776526678}{4736311618921872918919} a + \frac{2280037855100816322205}{4736311618921872918919}$

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:  \( 16088691144.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5^2:C_4$ (as 20T93):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $D_5^2:C_4$
Character table for $D_5^2:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 10.10.9627168800000.1

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 R $20$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
3469Data not computed