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Properties

Label 20.0.41931483557...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 5^{15}\cdot 13^{15}$
Root discriminant $60.41$
Ramified primes $2, 5, 13$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7425860, 0, 14851720, 0, 2170636, 0, 456976, 0, 114244, 0, 2197, 0, 17069, 0, 1014, 0, 286, 0, 13, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 13*x^18 + 286*x^16 + 1014*x^14 + 17069*x^12 + 2197*x^10 + 114244*x^8 + 456976*x^6 + 2170636*x^4 + 14851720*x^2 + 7425860)
 
gp: K = bnfinit(x^20 + 13*x^18 + 286*x^16 + 1014*x^14 + 17069*x^12 + 2197*x^10 + 114244*x^8 + 456976*x^6 + 2170636*x^4 + 14851720*x^2 + 7425860, 1)
 

Normalized defining polynomial

\( x^{20} + 13 x^{18} + 286 x^{16} + 1014 x^{14} + 17069 x^{12} + 2197 x^{10} + 114244 x^{8} + 456976 x^{6} + 2170636 x^{4} + 14851720 x^{2} + 7425860 \)

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:  \(419314835571431481344000000000000000=2^{28}\cdot 5^{15}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.41$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{13} a^{4}$, $\frac{1}{13} a^{5}$, $\frac{1}{13} a^{6}$, $\frac{1}{13} a^{7}$, $\frac{1}{338} a^{8} - \frac{1}{26} a^{7} - \frac{1}{26} a^{6} - \frac{1}{26} a^{5}$, $\frac{1}{338} a^{9} - \frac{1}{26} a^{5}$, $\frac{1}{338} a^{10} - \frac{1}{26} a^{6}$, $\frac{1}{338} a^{11} - \frac{1}{26} a^{7}$, $\frac{1}{4394} a^{12} - \frac{1}{26} a^{7} - \frac{1}{26} a^{6} - \frac{1}{26} a^{5}$, $\frac{1}{8788} a^{13} - \frac{1}{8788} a^{12} - \frac{1}{676} a^{9} - \frac{1}{676} a^{8} - \frac{1}{26} a^{6} - \frac{1}{26} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{26364} a^{14} + \frac{1}{26364} a^{12} + \frac{1}{2028} a^{10} + \frac{1}{2028} a^{8} - \frac{1}{26} a^{7} - \frac{1}{26} a^{6} + \frac{1}{78} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{26364} a^{15} + \frac{1}{26364} a^{13} + \frac{1}{2028} a^{11} + \frac{1}{2028} a^{9} - \frac{1}{26} a^{6} - \frac{1}{39} a^{5} - \frac{1}{26} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{1028196} a^{16} - \frac{1}{13182} a^{12} - \frac{1}{1014} a^{10} + \frac{5}{6084} a^{8} - \frac{4}{117} a^{6} - \frac{1}{26} a^{5} - \frac{1}{78} a^{4} - \frac{1}{2} a^{3} - \frac{4}{9} a^{2} - \frac{4}{9}$, $\frac{1}{1028196} a^{17} + \frac{1}{26364} a^{13} - \frac{1}{8788} a^{12} - \frac{1}{1014} a^{11} - \frac{1}{1521} a^{9} - \frac{1}{676} a^{8} - \frac{4}{117} a^{7} + \frac{1}{39} a^{5} - \frac{1}{26} a^{4} + \frac{1}{18} a^{3} - \frac{4}{9} a$, $\frac{1}{4525022223022392} a^{18} - \frac{883332245}{2262511111511196} a^{16} + \frac{617593331}{58013105423364} a^{14} - \frac{397693271}{9668850903894} a^{12} + \frac{1891271021}{2059636878936} a^{10} + \frac{5795582473}{4462546571028} a^{8} - \frac{1}{26} a^{7} + \frac{409569341}{257454609867} a^{6} + \frac{1857864659}{73558459962} a^{4} - \frac{1}{2} a^{3} - \frac{2159352065}{13202800506} a^{2} - \frac{560221409}{39608401518}$, $\frac{1}{4525022223022392} a^{19} - \frac{883332245}{2262511111511196} a^{17} + \frac{617593331}{58013105423364} a^{15} - \frac{397693271}{9668850903894} a^{13} + \frac{1891271021}{2059636878936} a^{11} + \frac{5795582473}{4462546571028} a^{9} - \frac{18985062077}{514909219734} a^{7} - \frac{1}{26} a^{6} - \frac{485653439}{36779229981} a^{5} - \frac{1}{26} a^{4} - \frac{2159352065}{13202800506} a^{3} - \frac{560221409}{39608401518} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( -1 \) (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:  \( 679419183.0669895 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{65}) \), 4.0.4394000.1, 5.1.338000.1, 10.2.7425860000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ $20$

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$2.10.14.5$x^{10} - 2 x^{5} - 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
2.10.14.5$x^{10} - 2 x^{5} - 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
5Data not computed
13Data not computed