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Properties

Label 22.6.61684606996...8125.2
Degree $22$
Signature $[6, 8]$
Discriminant $5^{11}\cdot 1831^{8}$
Root discriminant $34.35$
Ramified primes $5, 1831$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T13

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -4, 7, 4, 85, -56, 102, -459, -90, -523, -24, 64, 72, 218, 173, 272, -5, 142, -14, 14, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 5*x^20 + 14*x^19 - 14*x^18 + 142*x^17 - 5*x^16 + 272*x^15 + 173*x^14 + 218*x^13 + 72*x^12 + 64*x^11 - 24*x^10 - 523*x^9 - 90*x^8 - 459*x^7 + 102*x^6 - 56*x^5 + 85*x^4 + 4*x^3 + 7*x^2 - 4*x - 1)
 
gp: K = bnfinit(x^22 - 2*x^21 + 5*x^20 + 14*x^19 - 14*x^18 + 142*x^17 - 5*x^16 + 272*x^15 + 173*x^14 + 218*x^13 + 72*x^12 + 64*x^11 - 24*x^10 - 523*x^9 - 90*x^8 - 459*x^7 + 102*x^6 - 56*x^5 + 85*x^4 + 4*x^3 + 7*x^2 - 4*x - 1, 1)
 

Normalized defining polynomial

\( x^{22} - 2 x^{21} + 5 x^{20} + 14 x^{19} - 14 x^{18} + 142 x^{17} - 5 x^{16} + 272 x^{15} + 173 x^{14} + 218 x^{13} + 72 x^{12} + 64 x^{11} - 24 x^{10} - 523 x^{9} - 90 x^{8} - 459 x^{7} + 102 x^{6} - 56 x^{5} + 85 x^{4} + 4 x^{3} + 7 x^{2} - 4 x - 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6168460699614248235671671923828125=5^{11}\cdot 1831^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.35$
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:  $5, 1831$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{47} a^{20} + \frac{12}{47} a^{19} - \frac{7}{47} a^{18} + \frac{12}{47} a^{17} + \frac{4}{47} a^{16} + \frac{12}{47} a^{15} + \frac{7}{47} a^{14} - \frac{4}{47} a^{13} - \frac{15}{47} a^{12} + \frac{23}{47} a^{11} - \frac{8}{47} a^{10} - \frac{5}{47} a^{9} - \frac{17}{47} a^{8} - \frac{2}{47} a^{7} - \frac{19}{47} a^{6} + \frac{11}{47} a^{5} + \frac{10}{47} a^{4} - \frac{16}{47} a^{3} - \frac{12}{47} a^{2} - \frac{10}{47} a + \frac{6}{47}$, $\frac{1}{101581987079819159741470267} a^{21} + \frac{383962489179151977598761}{101581987079819159741470267} a^{20} - \frac{33023377970524802639988353}{101581987079819159741470267} a^{19} + \frac{33889026890443552308413869}{101581987079819159741470267} a^{18} + \frac{22654489864168760744877631}{101581987079819159741470267} a^{17} - \frac{38185108264066552545843402}{101581987079819159741470267} a^{16} + \frac{3165538680794952814968734}{101581987079819159741470267} a^{15} - \frac{36041572542037962953121919}{101581987079819159741470267} a^{14} + \frac{42576954608990110084476203}{101581987079819159741470267} a^{13} + \frac{15710103842316891145999739}{101581987079819159741470267} a^{12} - \frac{16289626381383339937472687}{101581987079819159741470267} a^{11} - \frac{5496122526804980434940662}{101581987079819159741470267} a^{10} + \frac{37162512315119568454840333}{101581987079819159741470267} a^{9} - \frac{3540368379286097421618202}{101581987079819159741470267} a^{8} + \frac{15311997519859427907881628}{101581987079819159741470267} a^{7} - \frac{13068995704834252600688603}{101581987079819159741470267} a^{6} + \frac{33104028772245654734585148}{101581987079819159741470267} a^{5} + \frac{10542750634584345762230533}{101581987079819159741470267} a^{4} + \frac{32552623596314552708376259}{101581987079819159741470267} a^{3} - \frac{29323902608197723902278591}{101581987079819159741470267} a^{2} + \frac{23482558155172561748135134}{101581987079819159741470267} a - \frac{44864396837923022862842708}{101581987079819159741470267}$

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:  $13$
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:  \( 185058540.947 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T13:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1320
The 16 conjugacy class representatives for t22n13
Character table for t22n13

Intermediate fields

\(\Q(\sqrt{5}) \), 11.3.11239665258721.1

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

Sibling fields

Degree 24 sibling: data not computed
Arithmetically equvalently 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 $22$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
5Data not computed
1831Data not computed