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Properties

Label 22.22.1158360636...3125.1
Degree $22$
Signature $[22, 0]$
Discriminant $5^{11}\cdot 421^{2}\cdot 115692385433^{2}$
Root discriminant $39.25$
Ramified primes $5, 421, 115692385433$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T47

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -13, 118, 425, -1613, -4249, 7336, 15763, -17000, -28981, 23007, 29509, -18923, -17186, 9370, 5647, -2683, -1001, 421, 88, -33, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 3*x^21 - 33*x^20 + 88*x^19 + 421*x^18 - 1001*x^17 - 2683*x^16 + 5647*x^15 + 9370*x^14 - 17186*x^13 - 18923*x^12 + 29509*x^11 + 23007*x^10 - 28981*x^9 - 17000*x^8 + 15763*x^7 + 7336*x^6 - 4249*x^5 - 1613*x^4 + 425*x^3 + 118*x^2 - 13*x - 1)
 
gp: K = bnfinit(x^22 - 3*x^21 - 33*x^20 + 88*x^19 + 421*x^18 - 1001*x^17 - 2683*x^16 + 5647*x^15 + 9370*x^14 - 17186*x^13 - 18923*x^12 + 29509*x^11 + 23007*x^10 - 28981*x^9 - 17000*x^8 + 15763*x^7 + 7336*x^6 - 4249*x^5 - 1613*x^4 + 425*x^3 + 118*x^2 - 13*x - 1, 1)
 

Normalized defining polynomial

\( x^{22} - 3 x^{21} - 33 x^{20} + 88 x^{19} + 421 x^{18} - 1001 x^{17} - 2683 x^{16} + 5647 x^{15} + 9370 x^{14} - 17186 x^{13} - 18923 x^{12} + 29509 x^{11} + 23007 x^{10} - 28981 x^{9} - 17000 x^{8} + 15763 x^{7} + 7336 x^{6} - 4249 x^{5} - 1613 x^{4} + 425 x^{3} + 118 x^{2} - 13 x - 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(115836063662590130514138078564453125=5^{11}\cdot 421^{2}\cdot 115692385433^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.25$
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:  $5, 421, 115692385433$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2227025078236132365707} a^{21} + \frac{845151641667628052296}{2227025078236132365707} a^{20} + \frac{967696812774636417910}{2227025078236132365707} a^{19} + \frac{384839328226069638755}{2227025078236132365707} a^{18} + \frac{587740798647621206833}{2227025078236132365707} a^{17} + \frac{444412456798246440642}{2227025078236132365707} a^{16} - \frac{816442154736135423577}{2227025078236132365707} a^{15} - \frac{979034667915737570914}{2227025078236132365707} a^{14} + \frac{545695828424019326276}{2227025078236132365707} a^{13} - \frac{876821745258557449269}{2227025078236132365707} a^{12} - \frac{46393845286969124826}{2227025078236132365707} a^{11} - \frac{205199352856578442063}{2227025078236132365707} a^{10} + \frac{383512345450104675207}{2227025078236132365707} a^{9} - \frac{957987735617915514783}{2227025078236132365707} a^{8} + \frac{647986235511005935113}{2227025078236132365707} a^{7} - \frac{171873004766058317395}{2227025078236132365707} a^{6} + \frac{1009006572829134136669}{2227025078236132365707} a^{5} - \frac{175849698365348399130}{2227025078236132365707} a^{4} + \frac{887693765904783797938}{2227025078236132365707} a^{3} + \frac{1101711933448412123576}{2227025078236132365707} a^{2} - \frac{853405891773441086708}{2227025078236132365707} a - \frac{604997647763735441895}{2227025078236132365707}$

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:  $21$
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:  \( 25661717777.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T47:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 79833600
The 112 conjugacy class representatives for t22n47 are not computed
Character table for t22n47 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 11.11.48706494267293.1

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
Degree 44 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R $22$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ $22$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
421Data not computed
115692385433Data not computed