# 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

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)

## Normalizeddefining 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

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

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

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 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$$1010T14[2, 2, 2, 2, 2]^{5} 2.12.12.1x^{12} - 48 x^{10} + 49 x^{8} + 8 x^{6} + 19 x^{4} - 24 x^{2} + 59$$2$$6$$12$12T134$[2, 2, 2, 2, 2, 2]^{6}$
12917Data not computed
459847Data not computed