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Properties

Label 20.0.15477458825...5184.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 3^{10}\cdot 13^{4}\cdot 43^{4}$
Root discriminant $16.20$
Ramified primes $2, 3, 13, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1013

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 5, 0, 25, 0, 65, 0, 81, 0, 41, 0, -1, 0, -20, 0, -7, 0, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 2*x^18 - 7*x^16 - 20*x^14 - x^12 + 41*x^10 + 81*x^8 + 65*x^6 + 25*x^4 + 5*x^2 + 1)
 
gp: K = bnfinit(x^20 + 2*x^18 - 7*x^16 - 20*x^14 - x^12 + 41*x^10 + 81*x^8 + 65*x^6 + 25*x^4 + 5*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 2 x^{18} - 7 x^{16} - 20 x^{14} - x^{12} + 41 x^{10} + 81 x^{8} + 65 x^{6} + 25 x^{4} + 5 x^{2} + 1 \)

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:  \(1547745882534914172125184=2^{28}\cdot 3^{10}\cdot 13^{4}\cdot 43^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.20$
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:  $2, 3, 13, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{16} - \frac{1}{6} a^{14} - \frac{1}{6} a^{12} + \frac{1}{3} a^{8} - \frac{1}{6} a^{6} + \frac{1}{3} a^{2} + \frac{1}{6}$, $\frac{1}{6} a^{17} - \frac{1}{6} a^{15} - \frac{1}{6} a^{13} + \frac{1}{3} a^{9} - \frac{1}{6} a^{7} + \frac{1}{3} a^{3} + \frac{1}{6} a$, $\frac{1}{10212} a^{18} - \frac{1}{12} a^{17} - \frac{17}{1702} a^{16} - \frac{1}{12} a^{15} + \frac{310}{2553} a^{14} + \frac{1}{12} a^{13} + \frac{1223}{10212} a^{12} + \frac{109}{851} a^{10} + \frac{1}{3} a^{9} - \frac{3235}{10212} a^{8} - \frac{5}{12} a^{7} + \frac{2929}{10212} a^{6} + \frac{665}{2553} a^{4} + \frac{2513}{10212} a^{2} + \frac{1}{4} a - \frac{881}{3404}$, $\frac{1}{10212} a^{19} + \frac{749}{10212} a^{17} - \frac{1}{12} a^{16} - \frac{1313}{10212} a^{15} - \frac{1}{12} a^{14} + \frac{31}{851} a^{13} + \frac{1}{12} a^{12} + \frac{109}{851} a^{11} + \frac{1191}{3404} a^{9} + \frac{1}{3} a^{8} - \frac{757}{2553} a^{7} - \frac{5}{12} a^{6} + \frac{665}{2553} a^{5} - \frac{4295}{10212} a^{3} + \frac{403}{2553} a + \frac{1}{4}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{3}{23} a^{18} + \frac{2}{69} a^{16} - \frac{29}{23} a^{14} - \frac{56}{69} a^{12} + \frac{83}{23} a^{10} + \frac{256}{69} a^{8} + \frac{118}{69} a^{6} - \frac{139}{23} a^{4} - \frac{268}{69} a^{2} - \frac{5}{69} \) (order $6$)
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:  \( 19197.4977797 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1013:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3686400
The 114 conjugacy class representatives for t20n1013 are not computed
Character table for t20n1013 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 10.0.4859704512.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 32 sibling: 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 R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$2.8.16.10$x^{8} + 2 x^{6} + 4 x^{5} + 6 x^{4} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
2.12.12.9$x^{12} - 18 x^{10} + 7 x^{8} - 28 x^{6} - x^{4} - 18 x^{2} - 7$$2$$6$$12$12T58$[2, 2, 2, 2]^{6}$
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.6.4.2$x^{6} - 13 x^{3} + 338$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.5.0.1$x^{5} - x + 10$$1$$5$$0$$C_5$$[\ ]^{5}$
43.5.0.1$x^{5} - x + 10$$1$$5$$0$$C_5$$[\ ]^{5}$