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Properties

Label 20.4.16746787578...0000.3
Degree $20$
Signature $[4, 8]$
Discriminant $2^{20}\cdot 5^{27}\cdot 11^{8}$
Root discriminant $45.83$
Ramified primes $2, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T534

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![125, 0, 2500, 0, 14125, 0, 16125, 0, 2625, 0, -3950, 0, -1650, 0, 50, 0, 120, 0, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 20*x^18 + 120*x^16 + 50*x^14 - 1650*x^12 - 3950*x^10 + 2625*x^8 + 16125*x^6 + 14125*x^4 + 2500*x^2 + 125)
 
gp: K = bnfinit(x^20 + 20*x^18 + 120*x^16 + 50*x^14 - 1650*x^12 - 3950*x^10 + 2625*x^8 + 16125*x^6 + 14125*x^4 + 2500*x^2 + 125, 1)
 

Normalized defining polynomial

\( x^{20} + 20 x^{18} + 120 x^{16} + 50 x^{14} - 1650 x^{12} - 3950 x^{10} + 2625 x^{8} + 16125 x^{6} + 14125 x^{4} + 2500 x^{2} + 125 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1674678757812500000000000000000000=2^{20}\cdot 5^{27}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.83$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{5} a^{8}$, $\frac{1}{5} a^{9}$, $\frac{1}{5} a^{10}$, $\frac{1}{5} a^{11}$, $\frac{1}{5} a^{12}$, $\frac{1}{5} a^{13}$, $\frac{1}{25} a^{14}$, $\frac{1}{25} a^{15}$, $\frac{1}{6325} a^{16} - \frac{14}{6325} a^{14} - \frac{119}{1265} a^{12} - \frac{98}{1265} a^{10} + \frac{3}{253} a^{8} + \frac{10}{253} a^{6} - \frac{13}{253} a^{4} + \frac{56}{253} a^{2} - \frac{24}{253}$, $\frac{1}{6325} a^{17} - \frac{14}{6325} a^{15} - \frac{119}{1265} a^{13} - \frac{98}{1265} a^{11} + \frac{3}{253} a^{9} + \frac{10}{253} a^{7} - \frac{13}{253} a^{5} + \frac{56}{253} a^{3} - \frac{24}{253} a$, $\frac{1}{83483675} a^{18} + \frac{6033}{83483675} a^{16} - \frac{1274606}{83483675} a^{14} + \frac{174943}{3339347} a^{12} + \frac{730599}{16696735} a^{10} - \frac{1436859}{16696735} a^{8} + \frac{419}{49841} a^{6} + \frac{331558}{3339347} a^{4} + \frac{689013}{3339347} a^{2} - \frac{170428}{3339347}$, $\frac{1}{83483675} a^{19} + \frac{6033}{83483675} a^{17} - \frac{1274606}{83483675} a^{15} + \frac{174943}{3339347} a^{13} + \frac{730599}{16696735} a^{11} - \frac{1436859}{16696735} a^{9} + \frac{419}{49841} a^{7} + \frac{331558}{3339347} a^{5} + \frac{689013}{3339347} a^{3} - \frac{170428}{3339347} a$

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:  $11$
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:  \( 671705048.729 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T534:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 25600
The 88 conjugacy class representatives for t20n534 are not computed
Character table for t20n534 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.17872314453125.1

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 $20$ R $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$11$11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$