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Properties

Label 20.10.1611283824...0000.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{46}\cdot 5^{20}\cdot 7^{4}$
Root discriminant $36.34$
Ramified primes $2, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T925

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, 0, 20, 0, 15, 0, -260, 0, 640, 0, -778, 0, 570, 0, -260, 0, 65, 0, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 + 65*x^16 - 260*x^14 + 570*x^12 - 778*x^10 + 640*x^8 - 260*x^6 + 15*x^4 + 20*x^2 - 4)
 
gp: K = bnfinit(x^20 - 10*x^18 + 65*x^16 - 260*x^14 + 570*x^12 - 778*x^10 + 640*x^8 - 260*x^6 + 15*x^4 + 20*x^2 - 4, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{18} + 65 x^{16} - 260 x^{14} + 570 x^{12} - 778 x^{10} + 640 x^{8} - 260 x^{6} + 15 x^{4} + 20 x^{2} - 4 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-16112838246400000000000000000000=-\,2^{46}\cdot 5^{20}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.34$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{5}$, $\frac{1}{35} a^{16} - \frac{2}{35} a^{14} - \frac{2}{35} a^{12} + \frac{1}{35} a^{10} + \frac{3}{7} a^{8} - \frac{9}{35} a^{6} + \frac{13}{35} a^{4} - \frac{12}{35} a^{2} - \frac{9}{35}$, $\frac{1}{35} a^{17} - \frac{2}{35} a^{15} - \frac{2}{35} a^{13} + \frac{1}{35} a^{11} + \frac{3}{7} a^{9} - \frac{9}{35} a^{7} + \frac{13}{35} a^{5} - \frac{12}{35} a^{3} - \frac{9}{35} a$, $\frac{1}{560} a^{18} - \frac{1}{70} a^{17} - \frac{1}{70} a^{16} + \frac{1}{35} a^{15} + \frac{17}{560} a^{14} - \frac{1}{14} a^{13} - \frac{5}{56} a^{12} + \frac{3}{35} a^{11} - \frac{13}{280} a^{10} + \frac{2}{7} a^{9} + \frac{73}{280} a^{8} - \frac{13}{35} a^{7} - \frac{27}{140} a^{6} + \frac{11}{35} a^{5} - \frac{47}{140} a^{4} - \frac{3}{7} a^{3} - \frac{3}{16} a^{2} - \frac{19}{70} a + \frac{97}{280}$, $\frac{1}{1120} a^{19} - \frac{1}{140} a^{17} - \frac{1}{70} a^{16} + \frac{17}{1120} a^{15} + \frac{1}{35} a^{14} + \frac{31}{560} a^{13} - \frac{1}{14} a^{12} + \frac{43}{560} a^{11} + \frac{3}{35} a^{10} - \frac{207}{560} a^{9} + \frac{2}{7} a^{8} + \frac{113}{280} a^{7} - \frac{13}{35} a^{6} - \frac{47}{280} a^{5} + \frac{11}{35} a^{4} - \frac{79}{160} a^{3} - \frac{3}{7} a^{2} - \frac{127}{560} a - \frac{19}{70}$

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:  $14$
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:  \( 780658452.338 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T925:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 409600
The 190 conjugacy class representatives for t20n925 are not computed
Character table for t20n925 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.6.15680000000000.1

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 R $20$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.10.10.9$x^{10} + 10 x^{8} + 10 x^{6} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 17$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
5.10.10.9$x^{10} + 10 x^{8} + 10 x^{6} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 17$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$