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Properties

Label 20.8.11043582468...7856.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{52}\cdot 31^{4}\cdot 227^{4}$
Root discriminant $35.66$
Ramified primes $2, 31, 227$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, 96, 0, -736, 208, 2256, -904, -3456, 1736, 2472, -1668, -272, 842, -344, -236, -8, -50, -4, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 2*x^18 - 4*x^17 - 50*x^16 - 8*x^15 - 236*x^14 - 344*x^13 + 842*x^12 - 272*x^11 - 1668*x^10 + 2472*x^9 + 1736*x^8 - 3456*x^7 - 904*x^6 + 2256*x^5 + 208*x^4 - 736*x^3 + 96*x - 8)
 
gp: K = bnfinit(x^20 + 2*x^18 - 4*x^17 - 50*x^16 - 8*x^15 - 236*x^14 - 344*x^13 + 842*x^12 - 272*x^11 - 1668*x^10 + 2472*x^9 + 1736*x^8 - 3456*x^7 - 904*x^6 + 2256*x^5 + 208*x^4 - 736*x^3 + 96*x - 8, 1)
 

Normalized defining polynomial

\( x^{20} + 2 x^{18} - 4 x^{17} - 50 x^{16} - 8 x^{15} - 236 x^{14} - 344 x^{13} + 842 x^{12} - 272 x^{11} - 1668 x^{10} + 2472 x^{9} + 1736 x^{8} - 3456 x^{7} - 904 x^{6} + 2256 x^{5} + 208 x^{4} - 736 x^{3} + 96 x - 8 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11043582468423783283165921017856=2^{52}\cdot 31^{4}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.66$
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:  $2, 31, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{16}$, $\frac{1}{4} a^{17}$, $\frac{1}{124} a^{18} - \frac{2}{31} a^{17} - \frac{7}{124} a^{16} - \frac{15}{124} a^{15} - \frac{2}{31} a^{14} + \frac{1}{31} a^{13} + \frac{3}{62} a^{12} - \frac{1}{62} a^{11} - \frac{7}{62} a^{10} - \frac{7}{62} a^{9} + \frac{6}{31} a^{8} - \frac{23}{62} a^{7} - \frac{5}{31} a^{6} + \frac{15}{31} a^{4} + \frac{10}{31} a^{3} - \frac{7}{31} a^{2} + \frac{10}{31} a - \frac{3}{31}$, $\frac{1}{18961350460378150803192} a^{19} - \frac{4579400024743603075}{4740337615094537700798} a^{18} + \frac{185573259016917467945}{3160225076729691800532} a^{17} - \frac{17079192798779846443}{249491453426028300042} a^{16} - \frac{585669350708586522251}{9480675230189075401596} a^{15} + \frac{146256342957425353231}{2370168807547268850399} a^{14} + \frac{1756098722364587405293}{9480675230189075401596} a^{13} + \frac{608204821065917452931}{4740337615094537700798} a^{12} - \frac{224698738706178504551}{3160225076729691800532} a^{11} - \frac{302579618330705882023}{2370168807547268850399} a^{10} + \frac{636998158995878965553}{4740337615094537700798} a^{9} - \frac{336846656158053076631}{4740337615094537700798} a^{8} + \frac{351229342692310629727}{4740337615094537700798} a^{7} - \frac{1171553204530963263557}{2370168807547268850399} a^{6} + \frac{600328312337323733203}{1580112538364845900266} a^{5} + \frac{70693139725094977853}{790056269182422950133} a^{4} - \frac{906640832292675949834}{2370168807547268850399} a^{3} - \frac{348021025321459567153}{2370168807547268850399} a^{2} + \frac{656830324712691006394}{2370168807547268850399} a - \frac{1017187123515859089613}{2370168807547268850399}$

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:  $13$
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:  \( 379314385.837 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.8.0.1}{8} }$ $16{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ $16{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ R $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.26.1$x^{8} + 4 x^{6} + 8 x^{3} + 8 x^{2} + 2$$8$$1$$26$$C_2^2:C_4$$[2, 3, 7/2, 4]$
2.12.26.27$x^{12} - 2 x^{10} + 4 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{3} + 2$$12$$1$$26$12T48$[4/3, 4/3, 2, 3]_{3}^{2}$
31Data not computed
227Data not computed