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Properties

Label 22.4.31964561289...7072.1
Degree $22$
Signature $[4, 9]$
Discriminant $-\,2^{33}\cdot 235475711\cdot 15802753811681$
Root discriminant $27.04$
Ramified primes $2, 235475711, 15802753811681$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T57

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 2, 1, 2, 2, -4, -1, -2, -8, 4, 0, -4, 13, -6, 2, 6, -10, 6, -3, -2, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 3*x^20 - 2*x^19 - 3*x^18 + 6*x^17 - 10*x^16 + 6*x^15 + 2*x^14 - 6*x^13 + 13*x^12 - 4*x^11 + 4*x^9 - 8*x^8 - 2*x^7 - x^6 - 4*x^5 + 2*x^4 + 2*x^3 + x^2 + 2*x - 1)
 
gp: K = bnfinit(x^22 - 2*x^21 + 3*x^20 - 2*x^19 - 3*x^18 + 6*x^17 - 10*x^16 + 6*x^15 + 2*x^14 - 6*x^13 + 13*x^12 - 4*x^11 + 4*x^9 - 8*x^8 - 2*x^7 - x^6 - 4*x^5 + 2*x^4 + 2*x^3 + x^2 + 2*x - 1, 1)
 

Normalized defining polynomial

\( x^{22} - 2 x^{21} + 3 x^{20} - 2 x^{19} - 3 x^{18} + 6 x^{17} - 10 x^{16} + 6 x^{15} + 2 x^{14} - 6 x^{13} + 13 x^{12} - 4 x^{11} + 4 x^{9} - 8 x^{8} - 2 x^{7} - x^{6} - 4 x^{5} + 2 x^{4} + 2 x^{3} + x^{2} + 2 x - 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-31964561289410824381582196867072=-\,2^{33}\cdot 235475711\cdot 15802753811681\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.04$
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:  $2, 235475711, 15802753811681$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{23} a^{21} + \frac{1}{23} a^{20} + \frac{6}{23} a^{19} - \frac{7}{23} a^{18} - \frac{1}{23} a^{17} + \frac{3}{23} a^{16} - \frac{1}{23} a^{15} + \frac{3}{23} a^{14} + \frac{11}{23} a^{13} + \frac{4}{23} a^{12} + \frac{2}{23} a^{11} + \frac{2}{23} a^{10} + \frac{6}{23} a^{9} - \frac{1}{23} a^{8} - \frac{11}{23} a^{7} + \frac{11}{23} a^{6} + \frac{9}{23} a^{5} + \frac{2}{23} a^{3} + \frac{8}{23} a^{2} + \frac{2}{23} a + \frac{8}{23}$

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:  $12$
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:  \( 12970191.905 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T57:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3186701844480000
The 1652 conjugacy class representatives for t22n57 are not computed
Character table for t22n57 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \)

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

Sibling fields

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 $20{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ $22$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }{,}\,{\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }{,}\,{\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ $22$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.10.0.1}{10} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
235475711Data not computed
15802753811681Data not computed