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Properties

Label 22.2.51543810007...2576.1
Degree $22$
Signature $[2, 10]$
Discriminant $2^{71}\cdot 3^{21}\cdot 211\cdot 337^{8}\cdot 310501^{8}\cdot 688136807609$
Root discriminant $97{,}032.48$
Ramified primes $2, 3, 211, 337, 310501, 688136807609$
Class number Not computed
Class group Not computed
Galois group 22T44

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3484724793731976, 0, -43296735359689596, 0, -4178116504369170, 0, 62534541123822, 0, 26575785014988, 0, 1677095605968, 0, 55486050798, 0, 1122151662, 0, 14371632, 0, 114074, 0, 513, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 513*x^20 + 114074*x^18 + 14371632*x^16 + 1122151662*x^14 + 55486050798*x^12 + 1677095605968*x^10 + 26575785014988*x^8 + 62534541123822*x^6 - 4178116504369170*x^4 - 43296735359689596*x^2 - 3484724793731976)
 
gp: K = bnfinit(x^22 + 513*x^20 + 114074*x^18 + 14371632*x^16 + 1122151662*x^14 + 55486050798*x^12 + 1677095605968*x^10 + 26575785014988*x^8 + 62534541123822*x^6 - 4178116504369170*x^4 - 43296735359689596*x^2 - 3484724793731976, 1)
 

Normalized defining polynomial

\( x^{22} + 513 x^{20} + 114074 x^{18} + 14371632 x^{16} + 1122151662 x^{14} + 55486050798 x^{12} + 1677095605968 x^{10} + 26575785014988 x^{8} + 62534541123822 x^{6} - 4178116504369170 x^{4} - 43296735359689596 x^{2} - 3484724793731976 \)

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:  \(51543810007530364950738948681743384761753476096510835701363261292919242474159040085359431452872546884912152576=2^{71}\cdot 3^{21}\cdot 211\cdot 337^{8}\cdot 310501^{8}\cdot 688136807609\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97{,}032.48$
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:  $2, 3, 211, 337, 310501, 688136807609$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{16}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{16}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{16}$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{20} - \frac{2950442336895763005424408194123522436283571976756718490815}{57887320607168480035725746532474501690526662399435613406516} a^{18} - \frac{7338001338100263739869009156974600085515755719442536959277}{28943660303584240017862873266237250845263331199717806703258} a^{16} - \frac{1863615119156654953980891776735522826638415204466559602944}{14471830151792120008931436633118625422631665599858903351629} a^{14} - \frac{1908865156040300215448588203615814517741870469194348054533}{28943660303584240017862873266237250845263331199717806703258} a^{12} + \frac{3308177665902484683773089588291819397024986636930560983303}{28943660303584240017862873266237250845263331199717806703258} a^{10} + \frac{746871812759802204132904074590997634538575333012148356024}{14471830151792120008931436633118625422631665599858903351629} a^{8} + \frac{6336870944220189766767509591463712821537141174725307648108}{14471830151792120008931436633118625422631665599858903351629} a^{6} - \frac{5902460395162287353805537079561292123575271969014728166243}{28943660303584240017862873266237250845263331199717806703258} a^{4} - \frac{4890199810199589231503791444038684133412319727949014822851}{28943660303584240017862873266237250845263331199717806703258} a^{2} + \frac{46449333229587174913088432383693176032339874284274981505}{14471830151792120008931436633118625422631665599858903351629}$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{21} - \frac{2950442336895763005424408194123522436283571976756718490815}{57887320607168480035725746532474501690526662399435613406516} a^{19} + \frac{3566914406845928134531213738072012668557954940208183196176}{14471830151792120008931436633118625422631665599858903351629} a^{17} - \frac{1}{2} a^{16} - \frac{1863615119156654953980891776735522826638415204466559602944}{14471830151792120008931436633118625422631665599858903351629} a^{15} - \frac{1908865156040300215448588203615814517741870469194348054533}{28943660303584240017862873266237250845263331199717806703258} a^{13} + \frac{3308177665902484683773089588291819397024986636930560983303}{28943660303584240017862873266237250845263331199717806703258} a^{11} + \frac{746871812759802204132904074590997634538575333012148356024}{14471830151792120008931436633118625422631665599858903351629} a^{9} + \frac{6336870944220189766767509591463712821537141174725307648108}{14471830151792120008931436633118625422631665599858903351629} a^{7} - \frac{5902460395162287353805537079561292123575271969014728166243}{28943660303584240017862873266237250845263331199717806703258} a^{5} - \frac{4890199810199589231503791444038684133412319727949014822851}{28943660303584240017862873266237250845263331199717806703258} a^{3} + \frac{46449333229587174913088432383693176032339874284274981505}{14471830151792120008931436633118625422631665599858903351629} a$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T44:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 16220160
The 104 conjugacy class representatives for t22n44 are not computed
Character table for t22n44 is not computed

Intermediate fields

11.11.118769262421915560193703211428553337469927424.1

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 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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
3Data not computed
211Data not computed
337Data not computed
310501Data not computed
688136807609Data not computed