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Properties

Label 20.0.70600893601...0416.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{59}\cdot 7^{5}\cdot 31^{4}\cdot 53^{4}$
Root discriminant $55.26$
Ramified primes $2, 7, 31, 53$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 20T174

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![134456, 0, 307328, 0, 153664, 0, 0, 0, 12656, 0, 12240, 0, 1808, 0, 0, 0, 56, 0, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 16*x^18 + 56*x^16 + 1808*x^12 + 12240*x^10 + 12656*x^8 + 153664*x^4 + 307328*x^2 + 134456)
 
gp: K = bnfinit(x^20 + 16*x^18 + 56*x^16 + 1808*x^12 + 12240*x^10 + 12656*x^8 + 153664*x^4 + 307328*x^2 + 134456, 1)
 

Normalized defining polynomial

\( x^{20} + 16 x^{18} + 56 x^{16} + 1808 x^{12} + 12240 x^{10} + 12656 x^{8} + 153664 x^{4} + 307328 x^{2} + 134456 \)

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:  \(70600893601130608587973468583100416=2^{59}\cdot 7^{5}\cdot 31^{4}\cdot 53^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.26$
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:  $2, 7, 31, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{14} a^{12} + \frac{1}{7} a^{10} + \frac{1}{7} a^{4} + \frac{2}{7} a^{2}$, $\frac{1}{14} a^{13} + \frac{1}{7} a^{11} + \frac{1}{7} a^{5} + \frac{2}{7} a^{3}$, $\frac{1}{196} a^{14} + \frac{1}{98} a^{12} + \frac{1}{7} a^{10} + \frac{11}{49} a^{6} + \frac{15}{49} a^{4} + \frac{2}{7} a^{2}$, $\frac{1}{196} a^{15} + \frac{1}{98} a^{13} + \frac{1}{7} a^{11} + \frac{11}{49} a^{7} + \frac{15}{49} a^{5} + \frac{2}{7} a^{3}$, $\frac{1}{1372} a^{16} + \frac{1}{686} a^{14} + \frac{1}{49} a^{12} + \frac{3}{14} a^{10} - \frac{125}{686} a^{8} + \frac{162}{343} a^{6} - \frac{19}{49} a^{4} + \frac{3}{7} a^{2}$, $\frac{1}{1372} a^{17} + \frac{1}{686} a^{15} + \frac{1}{49} a^{13} + \frac{3}{14} a^{11} - \frac{125}{686} a^{9} - \frac{19}{686} a^{7} - \frac{19}{49} a^{5} + \frac{3}{7} a^{3}$, $\frac{1}{449480900685169220} a^{18} + \frac{69494559081327}{449480900685169220} a^{16} + \frac{65723075411769}{64211557240738460} a^{14} + \frac{144041868193001}{4586539802909890} a^{12} - \frac{20003203554375541}{112370225171292305} a^{10} - \frac{13651099609840677}{224740450342584610} a^{8} - \frac{4732729325725146}{16052889310184615} a^{6} - \frac{70953697820598}{2293269901454945} a^{4} - \frac{125360461058352}{327609985922135} a^{2} + \frac{859907828401}{46801426560305}$, $\frac{1}{449480900685169220} a^{19} + \frac{69494559081327}{449480900685169220} a^{17} + \frac{65723075411769}{64211557240738460} a^{15} + \frac{144041868193001}{4586539802909890} a^{13} - \frac{20003203554375541}{112370225171292305} a^{11} - \frac{13651099609840677}{224740450342584610} a^{9} + \frac{6587430658734323}{32105778620369230} a^{7} - \frac{70953697820598}{2293269901454945} a^{5} - \frac{125360461058352}{327609985922135} a^{3} + \frac{859907828401}{46801426560305} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 629916041.473 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T174:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 960
The 35 conjugacy class representatives for t20n174
Character table for t20n174 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.14336.1, 5.3.26288.1, 10.6.1415288717312.1

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 24 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 $20$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31Data not computed
$53$53.8.4.1$x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
53.12.0.1$x^{12} - x + 12$$1$$12$$0$$C_{12}$$[\ ]^{12}$