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Properties

Label 22.0.10166877060...6736.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{32}\cdot 3^{28}\cdot 7^{4}\cdot 23^{4}\cdot 137^{16}$
Root discriminant $1000.75$
Ramified primes $2, 3, 7, 23, 137$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T46

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2916, 0, 648, 0, -11628, 0, 6588, 0, -1188, 0, -39240, 0, 19161, 0, 53361, 0, 19218, 0, 2082, 0, 81, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 81*x^20 + 2082*x^18 + 19218*x^16 + 53361*x^14 + 19161*x^12 - 39240*x^10 - 1188*x^8 + 6588*x^6 - 11628*x^4 + 648*x^2 + 2916)
 
gp: K = bnfinit(x^22 + 81*x^20 + 2082*x^18 + 19218*x^16 + 53361*x^14 + 19161*x^12 - 39240*x^10 - 1188*x^8 + 6588*x^6 - 11628*x^4 + 648*x^2 + 2916, 1)
 

Normalized defining polynomial

\( x^{22} + 81 x^{20} + 2082 x^{18} + 19218 x^{16} + 53361 x^{14} + 19161 x^{12} - 39240 x^{10} - 1188 x^{8} + 6588 x^{6} - 11628 x^{4} + 648 x^{2} + 2916 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1016687706043558313473747588024238284282042400008948883658412916736=-\,2^{32}\cdot 3^{28}\cdot 7^{4}\cdot 23^{4}\cdot 137^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1000.75$
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:  $2, 3, 7, 23, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{7}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{24} a^{16} - \frac{1}{24} a^{12} - \frac{1}{12} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{24} a^{17} - \frac{1}{24} a^{13} - \frac{1}{12} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2880} a^{18} - \frac{1}{48} a^{17} - \frac{19}{960} a^{16} - \frac{1}{24} a^{15} + \frac{3}{320} a^{14} - \frac{1}{48} a^{13} - \frac{17}{960} a^{12} - \frac{3}{80} a^{10} - \frac{1}{12} a^{9} + \frac{1}{80} a^{8} - \frac{1}{2} a^{7} + \frac{3}{16} a^{6} + \frac{1}{4} a^{5} - \frac{1}{20} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a + \frac{1}{80}$, $\frac{1}{8640} a^{19} + \frac{7}{960} a^{17} - \frac{1}{48} a^{16} - \frac{71}{2880} a^{15} - \frac{1}{24} a^{14} + \frac{103}{2880} a^{13} - \frac{1}{48} a^{12} - \frac{1}{80} a^{11} - \frac{19}{240} a^{9} - \frac{1}{4} a^{8} + \frac{19}{48} a^{7} - \frac{1}{2} a^{6} + \frac{3}{20} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{119}{240} a + \frac{1}{4}$, $\frac{1}{1493420671440000} a^{20} - \frac{6998383241}{82967815080000} a^{18} - \frac{1}{48} a^{17} + \frac{125709011017}{124451722620000} a^{16} - \frac{1}{24} a^{15} - \frac{564349644793}{248903445240000} a^{14} - \frac{1}{48} a^{13} + \frac{78815781607}{165935630160000} a^{12} + \frac{120142811489}{2592744221250} a^{10} - \frac{1}{12} a^{9} + \frac{2117417156077}{20741953770000} a^{8} - \frac{1}{2} a^{7} + \frac{3627844697647}{13827969180000} a^{6} + \frac{1}{4} a^{5} - \frac{1360561801883}{3456992295000} a^{4} + \frac{1}{4} a^{3} + \frac{7033477595401}{41483907540000} a^{2} + \frac{1}{4} a - \frac{826128757889}{4609323060000}$, $\frac{1}{8960524028640000} a^{21} - \frac{1}{2986841342880000} a^{20} + \frac{5452471471}{124451722620000} a^{19} - \frac{5452471471}{41483907540000} a^{18} + \frac{5696180886659}{1493420671440000} a^{17} - \frac{5696180886659}{497806890480000} a^{16} - \frac{4743208403917}{373355167860000} a^{15} + \frac{4743208403917}{124451722620000} a^{14} + \frac{31710295280857}{995613780960000} a^{13} - \frac{4054356920857}{331871260320000} a^{12} + \frac{183319225537}{124451722620000} a^{11} + \frac{3273673069463}{41483907540000} a^{10} + \frac{1188345789101}{62225861310000} a^{9} - \frac{1188345789101}{20741953770000} a^{8} - \frac{693395671103}{82967815080000} a^{7} - \frac{13134573508897}{27655938360000} a^{6} + \frac{195084730867}{20741953770000} a^{5} - \frac{195084730867}{6913984590000} a^{4} + \frac{27775431365401}{248903445240000} a^{3} + \frac{13708476174599}{82967815080000} a^{2} - \frac{12291819869639}{27655938360000} a + \frac{3073173749639}{9218646120000}$

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:  $10$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{4567075753}{14934206714400} a^{21} - \frac{42101919589}{1659356301600} a^{19} - \frac{3412551810019}{4978068904800} a^{17} - \frac{35527572977077}{4978068904800} a^{15} - \frac{23295128768543}{829678150800} a^{13} - \frac{16627560775277}{414839075400} a^{11} - \frac{3493734376577}{414839075400} a^{9} + \frac{698524944067}{69139845900} a^{7} - \frac{84087518498}{17284961475} a^{5} + \frac{837295383947}{414839075400} a^{3} + \frac{34699246297}{7682205100} a \) (order $4$)
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:  \( 89738500243100000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T46:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 39916800
The 62 conjugacy class representatives for t22n46 are not computed
Character table for t22n46 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 11.7.63019333158425674204677255696384.1

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

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ R $22$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ $22$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ $18{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.12.18.67$x^{12} + 2 x^{9} + 2 x^{7} + 2 x^{2} + 2$$12$$1$$18$$C_2 \times S_4$$[4/3, 4/3, 2]_{3}^{2}$
3Data not computed
7Data not computed
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.14.0.1$x^{14} - x + 7$$1$$14$$0$$C_{14}$$[\ ]^{14}$
$137$$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
137.5.4.1$x^{5} - 137$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
137.5.4.1$x^{5} - 137$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
137.5.4.1$x^{5} - 137$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
137.5.4.1$x^{5} - 137$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$