Properties

Label 22.2.147982356498821950109384704.2
Degree 22
Signature $[2, 10]$
Discriminant $2^{22}\cdot 12917^{2}\cdot 459847^{2}$
Ramified primes $2, 12917, 459847$
Class number 1 (GRH)
Class group Trivial (GRH)
Galois Group 22T51

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 0, 0, 1, 0, 1, 0, 3, 0, 0, 0, -1, 0, -2, 0, -2, 0, 1, 0, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + x^18 - 2*x^16 - 2*x^14 - x^12 + 3*x^8 + x^6 + x^4 - 1)
gp: K = bnfinit(x^22 + x^18 - 2*x^16 - 2*x^14 - x^12 + 3*x^8 + x^6 + x^4 - 1, 1)

Normalized defining polynomial

\(x^{22} \) \(\mathstrut +\mathstrut x^{18} \) \(\mathstrut -\mathstrut 2 x^{16} \) \(\mathstrut -\mathstrut 2 x^{14} \) \(\mathstrut -\mathstrut x^{12} \) \(\mathstrut +\mathstrut 3 x^{8} \) \(\mathstrut +\mathstrut x^{6} \) \(\mathstrut +\mathstrut x^{4} \) \(\mathstrut -\mathstrut 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[2, 10]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(147982356498821950109384704=2^{22}\cdot 12917^{2}\cdot 459847^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Ramified primes:  $2, 12917, 459847$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is not Galois over $\Q$.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$

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

Class group and class number

Trivial Abelian group, 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:  $11$
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:  \( a \),  \( 4 a^{21} - 4 a^{19} + 7 a^{17} - 14 a^{15} + 5 a^{13} - 6 a^{11} + 6 a^{9} + 6 a^{7} - 3 a^{5} + 5 a^{3} - 5 a \),  \( a^{20} + 2 a^{16} - 3 a^{14} - 4 a^{10} + a^{8} + 2 a^{6} + a^{4} + 2 a^{2} - 1 \),  \( 2 a^{20} - 2 a^{18} + 4 a^{16} - 8 a^{14} + 3 a^{12} - 5 a^{10} + 4 a^{8} + 4 a^{6} - a^{4} + 4 a^{2} - 4 \),  \( 6 a^{20} - 5 a^{18} + 10 a^{16} - 21 a^{14} + 5 a^{12} - 11 a^{10} + 10 a^{8} + 11 a^{6} - 2 a^{4} + 8 a^{2} - 8 \),  \( 5 a^{21} - 4 a^{19} + 9 a^{17} - 17 a^{15} + 4 a^{13} - 10 a^{11} + 6 a^{9} + 9 a^{7} - a^{5} + 8 a^{3} - 5 a - 1 \),  \( 3 a^{21} - 2 a^{20} - 3 a^{19} + 2 a^{18} + 5 a^{17} - 3 a^{16} - 11 a^{15} + 7 a^{14} + 4 a^{13} - 2 a^{12} - 4 a^{11} + 2 a^{10} + 6 a^{9} - 3 a^{8} + 5 a^{7} - 3 a^{6} - 3 a^{5} + 2 a^{4} + 3 a^{3} - 2 a^{2} - 4 a + 2 \),  \( 2 a^{21} + a^{20} - 2 a^{19} - 2 a^{18} + 3 a^{17} + 2 a^{16} - 7 a^{15} - 5 a^{14} + 2 a^{13} + 4 a^{12} - 2 a^{11} - a^{10} + 3 a^{9} + 3 a^{8} + 3 a^{7} + a^{6} - 2 a^{5} - 3 a^{4} + 2 a^{3} + 2 a^{2} - a - 2 \),  \( 5 a^{21} - 2 a^{20} - 4 a^{19} + a^{18} + 9 a^{17} - 4 a^{16} - 17 a^{15} + 6 a^{14} + 5 a^{13} - a^{12} - 10 a^{11} + 5 a^{10} + 7 a^{9} - a^{8} + 8 a^{7} - 4 a^{6} - 2 a^{5} + 8 a^{3} - 4 a^{2} - 6 a + 2 \),  \( 2 a^{21} + 2 a^{20} - a^{19} - 2 a^{18} + 4 a^{17} + 4 a^{16} - 6 a^{15} - 8 a^{14} + a^{13} + 3 a^{12} - 6 a^{11} - 5 a^{10} + a^{9} + 4 a^{8} + 3 a^{7} + 4 a^{6} + a^{5} + 5 a^{3} + 3 a^{2} - 2 a - 4 \),  \( a^{21} - 2 a^{20} + a^{18} + a^{17} - 3 a^{16} - 2 a^{15} + 6 a^{14} - a^{13} - a^{11} + 4 a^{10} + a^{9} - 2 a^{8} + 2 a^{7} - 3 a^{6} + a^{3} - 3 a^{2} + 1 \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 10254.8559154 \) (assuming GRH)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

22T51:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 40874803200
Conjugacy class representatives for 22T51
Character table for 22T51

Intermediate fields

11.1.5939843699.1

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

Sibling fields

Degree 22 sibling: 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 ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ $18{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.10.10.13$x^{10} - 15 x^{8} + 26 x^{6} - 22 x^{4} + 37 x^{2} - 59$$2$$5$$10$10T14$[2, 2, 2, 2, 2]^{5}$
2.12.12.1$x^{12} - 48 x^{10} + 49 x^{8} + 8 x^{6} + 19 x^{4} - 24 x^{2} + 59$$2$$6$$12$12T134$[2, 2, 2, 2, 2, 2]^{6}$
12917Data not computed
459847Data not computed