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Properties

Label 22.0.13329203788...9083.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,3^{11}\cdot 8674315276967^{2}$
Root discriminant $25.99$
Ramified primes $3, 8674315276967$
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, 2, 11, 12, 63, 67, 199, 155, 362, 323, 392, 369, 328, 258, 195, 111, 97, 20, 33, 1, 8, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 8*x^20 + x^19 + 33*x^18 + 20*x^17 + 97*x^16 + 111*x^15 + 195*x^14 + 258*x^13 + 328*x^12 + 369*x^11 + 392*x^10 + 323*x^9 + 362*x^8 + 155*x^7 + 199*x^6 + 67*x^5 + 63*x^4 + 12*x^3 + 11*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^22 - x^21 + 8*x^20 + x^19 + 33*x^18 + 20*x^17 + 97*x^16 + 111*x^15 + 195*x^14 + 258*x^13 + 328*x^12 + 369*x^11 + 392*x^10 + 323*x^9 + 362*x^8 + 155*x^7 + 199*x^6 + 67*x^5 + 63*x^4 + 12*x^3 + 11*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 8 x^{20} + x^{19} + 33 x^{18} + 20 x^{17} + 97 x^{16} + 111 x^{15} + 195 x^{14} + 258 x^{13} + 328 x^{12} + 369 x^{11} + 392 x^{10} + 323 x^{9} + 362 x^{8} + 155 x^{7} + 199 x^{6} + 67 x^{5} + 63 x^{4} + 12 x^{3} + 11 x^{2} + 2 x + 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-13329203788379546293009624459083=-\,3^{11}\cdot 8674315276967^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.99$
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:  $3, 8674315276967$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{115784657874210295} a^{21} + \frac{23431828034692631}{115784657874210295} a^{20} - \frac{7789772315321106}{23156931574842059} a^{19} + \frac{46662758512857011}{115784657874210295} a^{18} + \frac{2656249426351201}{23156931574842059} a^{17} + \frac{9254299551471028}{23156931574842059} a^{16} + \frac{28427075076587712}{115784657874210295} a^{15} + \frac{2147689549661762}{23156931574842059} a^{14} + \frac{2405882823249934}{23156931574842059} a^{13} + \frac{32929886603380953}{115784657874210295} a^{12} + \frac{22791518417790999}{115784657874210295} a^{11} - \frac{8590465991101053}{115784657874210295} a^{10} + \frac{9949809252096111}{115784657874210295} a^{9} + \frac{2372570600368855}{23156931574842059} a^{8} - \frac{42137702602895658}{115784657874210295} a^{7} - \frac{57212672410156026}{115784657874210295} a^{6} - \frac{4051106483033128}{115784657874210295} a^{5} - \frac{27750441681636164}{115784657874210295} a^{4} - \frac{10934051260486457}{23156931574842059} a^{3} - \frac{53125724354311553}{115784657874210295} a^{2} + \frac{11478490846526063}{23156931574842059} a + \frac{44118488733472137}{115784657874210295}$

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:  \( -\frac{43777949891663214}{115784657874210295} a^{21} + \frac{56176992000648426}{115784657874210295} a^{20} - \frac{70862937565902418}{23156931574842059} a^{19} + \frac{46304238082105751}{115784657874210295} a^{18} - \frac{273845407499128465}{23156931574842059} a^{17} - \frac{92101589895594629}{23156931574842059} a^{16} - \frac{3756014963871124433}{115784657874210295} a^{15} - \frac{704799598835654261}{23156931574842059} a^{14} - \frac{1298457807946225426}{23156931574842059} a^{13} - \frac{8109584073922414987}{115784657874210295} a^{12} - \frac{9965297578375354916}{115784657874210295} a^{11} - \frac{10502743477324340478}{115784657874210295} a^{10} - \frac{10686083960975967849}{115784657874210295} a^{9} - \frac{1451060702086977237}{23156931574842059} a^{8} - \frac{9823105793066203988}{115784657874210295} a^{7} - \frac{897754812270058486}{115784657874210295} a^{6} - \frac{4998718253209464818}{115784657874210295} a^{5} - \frac{66532980044445694}{115784657874210295} a^{4} - \frac{232323650375546467}{23156931574842059} a^{3} + \frac{563169515708121062}{115784657874210295} a^{2} - \frac{32570281122904994}{23156931574842059} a + \frac{110778693642358837}{115784657874210295} \) (order $6$)
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:  \( 6808139.27799 \) (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

\(\Q(\sqrt{-3}) \), 11.9.8674315276967.1

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 $22$ R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ $22$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ $18{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.14.0.1}{14} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
3Data not computed
8674315276967Data not computed