# Properties

 Label 22.2.147982356498821950109384704.1 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: K<a> := NumberField(PolynomialRing(Rationals())![-1, 0, 6, 0, -17, 0, 27, 0, -34, 0, 38, 0, -31, 0, 22, 0, -14, 0, 6, 0, -2, 0, 1]);
sage: K = NumberField(x^22 - 2*x^20 + 6*x^18 - 14*x^16 + 22*x^14 - 31*x^12 + 38*x^10 - 34*x^8 + 27*x^6 - 17*x^4 + 6*x^2 - 1,"a")
gp: K = bnfinit(x^22 - 2*x^20 + 6*x^18 - 14*x^16 + 22*x^14 - 31*x^12 + 38*x^10 - 34*x^8 + 27*x^6 - 17*x^4 + 6*x^2 - 1, 1)

## Normalizeddefining polynomial

$x^{22}$ $\mathstrut -\mathstrut 2 x^{20}$ $\mathstrut +\mathstrut 6 x^{18}$ $\mathstrut -\mathstrut 14 x^{16}$ $\mathstrut +\mathstrut 22 x^{14}$ $\mathstrut -\mathstrut 31 x^{12}$ $\mathstrut +\mathstrut 38 x^{10}$ $\mathstrut -\mathstrut 34 x^{8}$ $\mathstrut +\mathstrut 27 x^{6}$ $\mathstrut -\mathstrut 17 x^{4}$ $\mathstrut +\mathstrut 6 x^{2}$ $\mathstrut -\mathstrut 1$

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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{181} a^{20} + \frac{77}{181} a^{18} - \frac{65}{181} a^{16} - \frac{81}{181} a^{14} - \frac{42}{181} a^{12} + \frac{90}{181} a^{10} + \frac{89}{181} a^{8} - \frac{62}{181} a^{6} + \frac{16}{181} a^{4} - \frac{20}{181} a^{2} + \frac{55}{181}$, $\frac{1}{181} a^{21} + \frac{77}{181} a^{19} - \frac{65}{181} a^{17} - \frac{81}{181} a^{15} - \frac{42}{181} a^{13} + \frac{90}{181} a^{11} + \frac{89}{181} a^{9} - \frac{62}{181} a^{7} + \frac{16}{181} a^{5} - \frac{20}{181} a^{3} + \frac{55}{181} a$

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: $-1$ magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $\frac{346}{181} a^{20} - \frac{508}{181} a^{18} + \frac{1764}{181} a^{16} - \frac{3772}{181} a^{14} + \frac{5378}{181} a^{12} - \frac{7232}{181} a^{10} + \frac{8350}{181} a^{8} - \frac{6429}{181} a^{6} + \frac{4993}{181} a^{4} - \frac{2576}{181} a^{2} + \frac{568}{181}$,  $\frac{313}{181} a^{20} - \frac{515}{181} a^{18} + \frac{1737}{181} a^{16} - \frac{3814}{181} a^{14} + \frac{5678}{181} a^{12} - \frac{8030}{181} a^{10} + \frac{9395}{181} a^{8} - \frac{7641}{181} a^{6} + \frac{6094}{181} a^{4} - \frac{3364}{181} a^{2} + \frac{744}{181}$,  $a$,  $\frac{269}{181} a^{20} - \frac{464}{181} a^{18} + \frac{1520}{181} a^{16} - \frac{3327}{181} a^{14} + \frac{4992}{181} a^{12} - \frac{6922}{181} a^{10} + \frac{7832}{181} a^{8} - \frac{6180}{181} a^{6} + \frac{4666}{181} a^{4} - \frac{2122}{181} a^{2} + \frac{315}{181}$,  $\frac{421}{181} a^{21} - \frac{706}{181} a^{19} + \frac{2319}{181} a^{17} - \frac{5141}{181} a^{15} + \frac{7658}{181} a^{13} - \frac{10618}{181} a^{11} + \frac{12491}{181} a^{9} - \frac{10174}{181} a^{7} + \frac{7822}{181} a^{5} - \frac{4257}{181} a^{3} + \frac{1073}{181} a$,  $\frac{31}{181} a^{21} - \frac{147}{181} a^{19} + \frac{338}{181} a^{17} - \frac{882}{181} a^{15} + \frac{1775}{181} a^{13} - \frac{2459}{181} a^{11} + \frac{3302}{181} a^{9} - \frac{3551}{181} a^{7} + \frac{2668}{181} a^{5} - \frac{1887}{181} a^{3} + \frac{800}{181} a + 1$,  $\frac{257}{181} a^{21} + \frac{18}{181} a^{20} - \frac{302}{181} a^{19} - \frac{62}{181} a^{18} + \frac{1214}{181} a^{17} + \frac{97}{181} a^{16} - \frac{2536}{181} a^{15} - \frac{372}{181} a^{14} + \frac{3324}{181} a^{13} + \frac{511}{181} a^{12} - \frac{4744}{181} a^{11} - \frac{733}{181} a^{10} + \frac{5497}{181} a^{9} + \frac{1059}{181} a^{8} - \frac{3988}{181} a^{7} - \frac{1116}{181} a^{6} + \frac{3388}{181} a^{5} + \frac{831}{181} a^{4} - \frac{1701}{181} a^{3} - \frac{903}{181} a^{2} + \frac{198}{181} a + \frac{266}{181}$,  $\frac{421}{181} a^{21} - \frac{184}{181} a^{20} - \frac{706}{181} a^{19} + \frac{312}{181} a^{18} + \frac{2319}{181} a^{17} - \frac{1072}{181} a^{16} - \frac{5141}{181} a^{15} + \frac{2234}{181} a^{14} + \frac{7658}{181} a^{13} - \frac{3494}{181} a^{12} - \frac{10618}{181} a^{11} + \frac{4798}{181} a^{10} + \frac{12491}{181} a^{9} - \frac{5335}{181} a^{8} - \frac{10174}{181} a^{7} + \frac{4349}{181} a^{6} + \frac{7822}{181} a^{5} - \frac{3306}{181} a^{4} - \frac{4257}{181} a^{3} + \frac{1508}{181} a^{2} + \frac{892}{181} a - \frac{346}{181}$,  $\frac{42}{181} a^{21} - \frac{62}{181} a^{20} - \frac{205}{181} a^{19} + \frac{113}{181} a^{18} + \frac{347}{181} a^{17} - \frac{314}{181} a^{16} - \frac{1049}{181} a^{15} + \frac{678}{181} a^{14} + \frac{1856}{181} a^{13} - \frac{1016}{181} a^{12} - \frac{2193}{181} a^{11} + \frac{1117}{181} a^{10} + \frac{2833}{181} a^{9} - \frac{1174}{181} a^{8} - \frac{2785}{181} a^{7} + \frac{948}{181} a^{6} + \frac{1577}{181} a^{5} - \frac{630}{181} a^{4} - \frac{1202}{181} a^{3} + \frac{154}{181} a^{2} + \frac{500}{181} a - \frac{152}{181}$,  $\frac{305}{181} a^{21} - \frac{112}{181} a^{20} - \frac{407}{181} a^{19} + \frac{245}{181} a^{18} + \frac{1533}{181} a^{17} - \frac{684}{181} a^{16} - \frac{3166}{181} a^{15} + \frac{1651}{181} a^{14} + \frac{4385}{181} a^{13} - \frac{2536}{181} a^{12} - \frac{6035}{181} a^{11} + \frac{3495}{181} a^{10} + \frac{6692}{181} a^{9} - \frac{4176}{181} a^{8} - \frac{4792}{181} a^{7} + \frac{3324}{181} a^{6} + \frac{3794}{181} a^{5} - \frac{2335}{181} a^{4} - \frac{1756}{181} a^{3} + \frac{1335}{181} a^{2} + \frac{304}{181} a - \frac{187}{181}$,  $\frac{136}{181} a^{21} - \frac{44}{181} a^{20} - \frac{207}{181} a^{19} + \frac{51}{181} a^{18} + \frac{753}{181} a^{17} - \frac{217}{181} a^{16} - \frac{1604}{181} a^{15} + \frac{487}{181} a^{14} + \frac{2433}{181} a^{13} - \frac{686}{181} a^{12} - \frac{3507}{181} a^{11} + \frac{1108}{181} a^{10} + \frac{4140}{181} a^{9} - \frac{1563}{181} a^{8} - \frac{3545}{181} a^{7} + \frac{1461}{181} a^{6} + \frac{2900}{181} a^{5} - \frac{1428}{181} a^{4} - \frac{1453}{181} a^{3} + \frac{1061}{181} a^{2} + \frac{421}{181} a - \frac{429}{181}$ (assuming GRH) magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $11319.5749854$ (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 ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $18{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ $18{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.10.10.14$x^{10} + 5 x^{8} - 50 x^{6} - 58 x^{4} + 49 x^{2} + 21$$2$$5$$1010T14[2, 2, 2, 2, 2]^{5} 2.12.12.22x^{12} - 52 x^{10} - 7 x^{8} + 32 x^{6} + 35 x^{4} - 44 x^{2} - 29$$2$$6$$12$12T134$[2, 2, 2, 2, 2, 2]^{6}$
12917Data not computed
459847Data not computed