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Properties

Label 20.0.78699375727...7056.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{16}\cdot 11491^{4}$
Root discriminant $31.25$
Ramified primes $2, 3, 11491$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T794

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![441, 0, -1062, 0, 2145, 0, -2619, 0, 2517, 0, -1824, 0, 958, 0, -356, 0, 90, 0, -14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 14*x^18 + 90*x^16 - 356*x^14 + 958*x^12 - 1824*x^10 + 2517*x^8 - 2619*x^6 + 2145*x^4 - 1062*x^2 + 441)
 
gp: K = bnfinit(x^20 - 14*x^18 + 90*x^16 - 356*x^14 + 958*x^12 - 1824*x^10 + 2517*x^8 - 2619*x^6 + 2145*x^4 - 1062*x^2 + 441, 1)
 

Normalized defining polynomial

\( x^{20} - 14 x^{18} + 90 x^{16} - 356 x^{14} + 958 x^{12} - 1824 x^{10} + 2517 x^{8} - 2619 x^{6} + 2145 x^{4} - 1062 x^{2} + 441 \)

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:  \(786993757275194005675029037056=2^{20}\cdot 3^{16}\cdot 11491^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.25$
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:  $2, 3, 11491$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{14} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{15} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9}$, $\frac{1}{972109893} a^{18} - \frac{36957058}{324036631} a^{16} + \frac{205249715}{972109893} a^{14} - \frac{237156293}{972109893} a^{12} - \frac{112778324}{324036631} a^{10} - \frac{387925372}{972109893} a^{8} + \frac{35480995}{324036631} a^{6} - \frac{115653309}{324036631} a^{4} + \frac{10942762}{324036631} a^{2} - \frac{149171720}{324036631}$, $\frac{1}{6804769251} a^{19} - \frac{51570527}{324036631} a^{17} + \frac{205249715}{6804769251} a^{15} + \frac{2679173386}{6804769251} a^{13} - \frac{760851586}{2268256417} a^{11} + \frac{1556294414}{6804769251} a^{9} + \frac{359517626}{2268256417} a^{7} + \frac{532419953}{2268256417} a^{5} - \frac{637130500}{2268256417} a^{3} - \frac{1121281613}{2268256417} a$

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:  \( -\frac{9854}{1601499} a^{18} + \frac{81118}{1601499} a^{16} - \frac{91225}{533833} a^{14} + \frac{341941}{1601499} a^{12} + \frac{774691}{1601499} a^{10} - \frac{1338859}{533833} a^{8} + \frac{2597121}{533833} a^{6} - \frac{2470166}{533833} a^{4} + \frac{2153920}{533833} a^{2} - \frac{397435}{533833} \) (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:  \( 23021864.9911 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T794:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n794 are not computed
Character table for t20n794 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.310257.1, 10.0.288778218147.1

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 R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
11491Data not computed