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Properties

Label 22.4.12080573498...4063.1
Degree $22$
Signature $[4, 9]$
Discriminant $-\,13\cdot 9292748844744727822944291851$
Root discriminant $20.99$
Ramified primes $13, 9292748844744727822944291851$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T59

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 2, 2, -8, 10, -4, -14, 25, -16, -11, 35, -32, 2, 28, -31, 10, 11, -15, 6, 2, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 3*x^21 + 2*x^20 + 6*x^19 - 15*x^18 + 11*x^17 + 10*x^16 - 31*x^15 + 28*x^14 + 2*x^13 - 32*x^12 + 35*x^11 - 11*x^10 - 16*x^9 + 25*x^8 - 14*x^7 - 4*x^6 + 10*x^5 - 8*x^4 + 2*x^3 + 2*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^22 - 3*x^21 + 2*x^20 + 6*x^19 - 15*x^18 + 11*x^17 + 10*x^16 - 31*x^15 + 28*x^14 + 2*x^13 - 32*x^12 + 35*x^11 - 11*x^10 - 16*x^9 + 25*x^8 - 14*x^7 - 4*x^6 + 10*x^5 - 8*x^4 + 2*x^3 + 2*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - 3 x^{21} + 2 x^{20} + 6 x^{19} - 15 x^{18} + 11 x^{17} + 10 x^{16} - 31 x^{15} + 28 x^{14} + 2 x^{13} - 32 x^{12} + 35 x^{11} - 11 x^{10} - 16 x^{9} + 25 x^{8} - 14 x^{7} - 4 x^{6} + 10 x^{5} - 8 x^{4} + 2 x^{3} + 2 x^{2} - 2 x + 1 \)

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:  \(-120805734981681461698275794063=-\,13\cdot 9292748844744727822944291851\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.99$
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:  $13, 9292748844744727822944291851$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{149} a^{21} - \frac{30}{149} a^{20} + \frac{67}{149} a^{19} - \frac{15}{149} a^{18} - \frac{57}{149} a^{17} + \frac{60}{149} a^{16} + \frac{29}{149} a^{15} - \frac{69}{149} a^{14} - \frac{46}{149} a^{13} + \frac{52}{149} a^{12} + \frac{54}{149} a^{11} + \frac{67}{149} a^{10} - \frac{32}{149} a^{9} - \frac{46}{149} a^{8} - \frac{74}{149} a^{7} + \frac{47}{149} a^{6} + \frac{68}{149} a^{5} - \frac{38}{149} a^{4} - \frac{25}{149} a^{3} - \frac{68}{149} a^{2} + \frac{50}{149} a - \frac{11}{149}$

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:  \( 1008504.24777 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T59:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1124000727777607680000
The 1002 conjugacy class representatives for t22n59 are not computed
Character table for t22n59 is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.10.0.1}{10} }$ ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ $15{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.13.0.1}{13} }{,}\,{\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.13.0.1}{13} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R $22$ $21{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.13.0.1}{13} }{,}\,{\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.13.0.1}{13} }{,}\,{\href{/LocalNumberField/43.9.0.1}{9} }$ ${\href{/LocalNumberField/47.13.0.1}{13} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
13Data not computed
9292748844744727822944291851Data not computed