↑

Properties

Label 20.0.64000000000...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 5^{22}$
Root discriminant $15.50$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $C_2\times F_5$ (as 20T13)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -5, 0, 15, 0, -10, 0, -75, 0, 157, 0, -75, 0, -10, 0, 15, 0, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 + 15*x^16 - 10*x^14 - 75*x^12 + 157*x^10 - 75*x^8 - 10*x^6 + 15*x^4 - 5*x^2 + 1)
 
gp: K = bnfinit(x^20 - 5*x^18 + 15*x^16 - 10*x^14 - 75*x^12 + 157*x^10 - 75*x^8 - 10*x^6 + 15*x^4 - 5*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{18} + 15 x^{16} - 10 x^{14} - 75 x^{12} + 157 x^{10} - 75 x^{8} - 10 x^{6} + 15 x^{4} - 5 x^{2} + 1 \)

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:  \(640000000000000000000000=2^{28}\cdot 5^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.50$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{158} a^{16} - \frac{19}{79} a^{14} + \frac{2}{79} a^{12} - \frac{25}{158} a^{10} - \frac{1}{2} a^{9} - \frac{22}{79} a^{8} - \frac{25}{158} a^{6} - \frac{1}{2} a^{5} - \frac{75}{158} a^{4} + \frac{41}{158} a^{2} - \frac{1}{2} a - \frac{39}{79}$, $\frac{1}{158} a^{17} - \frac{19}{79} a^{15} + \frac{2}{79} a^{13} - \frac{25}{158} a^{11} + \frac{35}{158} a^{9} - \frac{1}{2} a^{8} - \frac{25}{158} a^{7} + \frac{2}{79} a^{5} - \frac{1}{2} a^{4} + \frac{41}{158} a^{3} + \frac{1}{158} a - \frac{1}{2}$, $\frac{1}{6636} a^{18} - \frac{1}{316} a^{17} - \frac{13}{6636} a^{16} + \frac{19}{158} a^{15} + \frac{17}{237} a^{14} - \frac{1}{79} a^{13} - \frac{397}{3318} a^{12} - \frac{27}{158} a^{11} + \frac{1069}{6636} a^{10} - \frac{35}{316} a^{9} - \frac{2389}{6636} a^{8} - \frac{27}{158} a^{7} - \frac{412}{1659} a^{6} + \frac{77}{158} a^{5} - \frac{26}{237} a^{4} - \frac{30}{79} a^{3} - \frac{2687}{6636} a^{2} + \frac{157}{316} a + \frac{1289}{6636}$, $\frac{1}{6636} a^{19} + \frac{2}{1659} a^{17} - \frac{1}{316} a^{16} - \frac{23}{474} a^{15} + \frac{19}{158} a^{14} - \frac{355}{3318} a^{13} - \frac{1}{79} a^{12} - \frac{1115}{6636} a^{11} - \frac{27}{158} a^{10} + \frac{416}{1659} a^{9} - \frac{35}{316} a^{8} + \frac{701}{1659} a^{7} - \frac{27}{158} a^{6} - \frac{23}{237} a^{5} + \frac{77}{158} a^{4} + \frac{3151}{6636} a^{3} - \frac{30}{79} a^{2} + \frac{655}{3318} a + \frac{157}{316}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{1585}{474} a^{19} + \frac{7225}{474} a^{17} - \frac{20633}{474} a^{15} + \frac{6935}{474} a^{13} + \frac{121385}{474} a^{11} - \frac{97630}{237} a^{9} + \frac{18110}{237} a^{7} + \frac{27461}{474} a^{5} - \frac{5885}{237} a^{3} + \frac{3085}{474} a \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 13353.721510432046 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(i, \sqrt{5})\), 5.1.50000.1, 10.0.800000000000.1, 10.0.160000000000.1, 10.2.12500000000.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$