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Properties

Label 20.8.30517578125...000.13
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 5^{35}$
Root discriminant $33.44$
Ramified primes $2, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T344

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, 0, 75, 0, 175, 0, -250, 0, -400, 0, 245, 0, 200, 0, -75, 0, -30, 0, 5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 5*x^18 - 30*x^16 - 75*x^14 + 200*x^12 + 245*x^10 - 400*x^8 - 250*x^6 + 175*x^4 + 75*x^2 + 5)
 
gp: K = bnfinit(x^20 + 5*x^18 - 30*x^16 - 75*x^14 + 200*x^12 + 245*x^10 - 400*x^8 - 250*x^6 + 175*x^4 + 75*x^2 + 5, 1)
 

Normalized defining polynomial

\( x^{20} + 5 x^{18} - 30 x^{16} - 75 x^{14} + 200 x^{12} + 245 x^{10} - 400 x^{8} - 250 x^{6} + 175 x^{4} + 75 x^{2} + 5 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3051757812500000000000000000000=2^{20}\cdot 5^{35}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.44$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{3}{7} a^{10} - \frac{2}{7} a^{8} - \frac{3}{7} a^{4} + \frac{3}{7} a^{2} - \frac{3}{7}$, $\frac{1}{7} a^{13} - \frac{3}{7} a^{11} - \frac{2}{7} a^{9} - \frac{3}{7} a^{5} + \frac{3}{7} a^{3} - \frac{3}{7} a$, $\frac{1}{7} a^{14} + \frac{3}{7} a^{10} + \frac{1}{7} a^{8} - \frac{3}{7} a^{6} + \frac{1}{7} a^{4} - \frac{1}{7} a^{2} - \frac{2}{7}$, $\frac{1}{7} a^{15} + \frac{3}{7} a^{11} + \frac{1}{7} a^{9} - \frac{3}{7} a^{7} + \frac{1}{7} a^{5} - \frac{1}{7} a^{3} - \frac{2}{7} a$, $\frac{1}{1043} a^{16} + \frac{72}{1043} a^{14} + \frac{4}{1043} a^{12} - \frac{199}{1043} a^{10} + \frac{487}{1043} a^{8} + \frac{2}{1043} a^{6} - \frac{401}{1043} a^{4} - \frac{43}{1043} a^{2} + \frac{1}{149}$, $\frac{1}{1043} a^{17} + \frac{72}{1043} a^{15} + \frac{4}{1043} a^{13} - \frac{199}{1043} a^{11} + \frac{487}{1043} a^{9} + \frac{2}{1043} a^{7} - \frac{401}{1043} a^{5} - \frac{43}{1043} a^{3} + \frac{1}{149} a$, $\frac{1}{1513393} a^{18} + \frac{389}{1513393} a^{16} - \frac{21872}{1513393} a^{14} + \frac{28485}{1513393} a^{12} - \frac{556382}{1513393} a^{10} - \frac{132891}{1513393} a^{8} + \frac{278267}{1513393} a^{6} + \frac{358133}{1513393} a^{4} + \frac{576416}{1513393} a^{2} - \frac{32349}{1513393}$, $\frac{1}{1513393} a^{19} + \frac{389}{1513393} a^{17} - \frac{21872}{1513393} a^{15} + \frac{28485}{1513393} a^{13} - \frac{556382}{1513393} a^{11} - \frac{132891}{1513393} a^{9} + \frac{278267}{1513393} a^{7} + \frac{358133}{1513393} a^{5} + \frac{576416}{1513393} a^{3} - \frac{32349}{1513393} a$

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:  $13$
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:  \( 46170606.15 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T344:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 80 conjugacy class representatives for t20n344 are not computed
Character table for t20n344 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.390625.1, \(\Q(\zeta_{25})^+\)

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 ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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