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Properties

Label 20.0.12020004674...2112.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 7^{15}\cdot 11^{9}$
Root discriminant $35.81$
Ramified primes $2, 7, 11$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![189314048, 0, 0, 0, 13522432, 0, 0, 0, 241472, 0, 21952, 0, 6272, 0, -784, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 784*x^14 + 6272*x^12 + 21952*x^10 + 241472*x^8 + 13522432*x^4 + 189314048)
 
gp: K = bnfinit(x^20 - 784*x^14 + 6272*x^12 + 21952*x^10 + 241472*x^8 + 13522432*x^4 + 189314048, 1)
 

Normalized defining polynomial

\( x^{20} - 784 x^{14} + 6272 x^{12} + 21952 x^{10} + 241472 x^{8} + 13522432 x^{4} + 189314048 \)

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:  \(12020004674398148709330936922112=2^{30}\cdot 7^{15}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.81$
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:  $2, 7, 11$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{28} a^{4}$, $\frac{1}{28} a^{5}$, $\frac{1}{56} a^{6}$, $\frac{1}{56} a^{7}$, $\frac{1}{784} a^{8}$, $\frac{1}{784} a^{9}$, $\frac{1}{6272} a^{10} - \frac{1}{112} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{6272} a^{11} - \frac{1}{112} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{87808} a^{12} + \frac{1}{224} a^{6} - \frac{1}{56} a^{4} - \frac{1}{8} a^{2}$, $\frac{1}{87808} a^{13} + \frac{1}{224} a^{7} - \frac{1}{56} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{175616} a^{14} - \frac{1}{3136} a^{8} - \frac{1}{112} a^{6} + \frac{1}{112} a^{4}$, $\frac{1}{175616} a^{15} - \frac{1}{3136} a^{9} - \frac{1}{112} a^{7} + \frac{1}{112} a^{5}$, $\frac{1}{2458624} a^{16} - \frac{1}{1568} a^{8} - \frac{1}{224} a^{6} + \frac{1}{112} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{2458624} a^{17} - \frac{1}{1568} a^{9} - \frac{1}{224} a^{7} + \frac{1}{112} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{14243757860864} a^{18} + \frac{643105}{3560939465216} a^{16} + \frac{418311}{508705637888} a^{14} + \frac{1136783}{254352818944} a^{12} + \frac{70447}{1297718464} a^{10} + \frac{1758431}{4542014624} a^{8} + \frac{113817}{20276851} a^{6} - \frac{1748983}{324429616} a^{4} - \frac{386215}{5793386} a^{2} - \frac{2321853}{5793386}$, $\frac{1}{14243757860864} a^{19} + \frac{643105}{3560939465216} a^{17} + \frac{418311}{508705637888} a^{15} + \frac{1136783}{254352818944} a^{13} + \frac{70447}{1297718464} a^{11} + \frac{1758431}{4542014624} a^{9} + \frac{113817}{20276851} a^{7} - \frac{1748983}{324429616} a^{5} - \frac{386215}{5793386} a^{3} - \frac{2321853}{5793386} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 3970499.4781347266 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.241472.2, 10.0.246071287.1

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 $20$ $20$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{10}$ $20$

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
$2$2.10.15.13$x^{10} - 2 x^{8} - 4 x^{6} - 48 x^{2} - 96$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
7Data not computed
$11$11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.10.9.8$x^{10} + 33$$10$$1$$9$$C_{10}$$[\ ]_{10}$