↑

Properties

Label 20.8.30389971566...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{12}\cdot 5^{10}\cdot 52501^{4}$
Root discriminant $29.79$
Ramified primes $2, 5, 52501$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T760

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 107, 0, -337, 0, -500, 0, 1080, 0, 1326, 0, 239, 0, -149, 0, -48, 0, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + x^18 - 48*x^16 - 149*x^14 + 239*x^12 + 1326*x^10 + 1080*x^8 - 500*x^6 - 337*x^4 + 107*x^2 + 1)
 
gp: K = bnfinit(x^20 + x^18 - 48*x^16 - 149*x^14 + 239*x^12 + 1326*x^10 + 1080*x^8 - 500*x^6 - 337*x^4 + 107*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} + x^{18} - 48 x^{16} - 149 x^{14} + 239 x^{12} + 1326 x^{10} + 1080 x^{8} - 500 x^{6} - 337 x^{4} + 107 x^{2} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(303899715661508400040000000000=2^{12}\cdot 5^{10}\cdot 52501^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.79$
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:  $2, 5, 52501$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{44} a^{16} - \frac{1}{4} a^{15} - \frac{5}{22} a^{14} - \frac{1}{4} a^{12} + \frac{9}{44} a^{10} + \frac{1}{4} a^{9} - \frac{3}{44} a^{8} + \frac{1}{4} a^{7} - \frac{1}{22} a^{6} - \frac{1}{4} a^{5} - \frac{5}{22} a^{4} + \frac{1}{4} a^{3} - \frac{13}{44} a^{2} - \frac{1}{2} a - \frac{3}{44}$, $\frac{1}{44} a^{17} + \frac{1}{44} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{9}{44} a^{11} - \frac{1}{2} a^{10} - \frac{7}{22} a^{9} - \frac{13}{44} a^{7} + \frac{1}{4} a^{6} + \frac{1}{44} a^{5} - \frac{1}{4} a^{4} + \frac{5}{11} a^{3} + \frac{1}{4} a^{2} + \frac{19}{44} a - \frac{1}{4}$, $\frac{1}{4073963476} a^{18} - \frac{2408026}{1018490869} a^{16} - \frac{1}{4} a^{15} + \frac{789089687}{4073963476} a^{14} - \frac{1}{4} a^{13} + \frac{13520556}{1018490869} a^{12} - \frac{1}{2} a^{11} + \frac{1864566149}{4073963476} a^{10} + \frac{1}{4} a^{9} - \frac{68146515}{2036981738} a^{8} - \frac{1}{2} a^{7} + \frac{1729478915}{4073963476} a^{6} - \frac{1}{2} a^{5} - \frac{380459987}{2036981738} a^{4} - \frac{1}{2} a^{3} + \frac{222715093}{1018490869} a^{2} + \frac{1}{4} a + \frac{851746827}{4073963476}$, $\frac{1}{4073963476} a^{19} - \frac{2408026}{1018490869} a^{17} - \frac{114700591}{2036981738} a^{15} - \frac{1}{4} a^{14} + \frac{13520556}{1018490869} a^{13} - \frac{1}{4} a^{12} + \frac{1864566149}{4073963476} a^{11} - \frac{1}{2} a^{10} + \frac{882197839}{4073963476} a^{9} + \frac{1}{4} a^{8} - \frac{331498423}{1018490869} a^{7} - \frac{1}{2} a^{6} - \frac{1779410843}{4073963476} a^{5} - \frac{1}{2} a^{4} + \frac{1909351241}{4073963476} a^{3} - \frac{1}{2} a^{2} - \frac{1185234911}{4073963476} a + \frac{1}{4}$

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:  \( 23159673.7028 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T760:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n760 are not computed
Character table for t20n760 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.8613609378125.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.7$x^{8} + 2 x^{6} + 4 x^{5} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
52501Data not computed