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Properties

Label 22.0.22378993818...5627.2
Degree $22$
Signature $[0, 11]$
Discriminant $-\,3^{11}\cdot 1831^{8}$
Root discriminant $26.61$
Ramified primes $3, 1831$
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, 13, -8, 11, -76, 42, 29, 130, -135, -76, -50, 188, -8, -19, -76, 31, 2, 18, -10, 1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + x^20 - 10*x^19 + 18*x^18 + 2*x^17 + 31*x^16 - 76*x^15 - 19*x^14 - 8*x^13 + 188*x^12 - 50*x^11 - 76*x^10 - 135*x^9 + 130*x^8 + 29*x^7 + 42*x^6 - 76*x^5 + 11*x^4 - 8*x^3 + 13*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^22 - 2*x^21 + x^20 - 10*x^19 + 18*x^18 + 2*x^17 + 31*x^16 - 76*x^15 - 19*x^14 - 8*x^13 + 188*x^12 - 50*x^11 - 76*x^10 - 135*x^9 + 130*x^8 + 29*x^7 + 42*x^6 - 76*x^5 + 11*x^4 - 8*x^3 + 13*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - 2 x^{21} + x^{20} - 10 x^{19} + 18 x^{18} + 2 x^{17} + 31 x^{16} - 76 x^{15} - 19 x^{14} - 8 x^{13} + 188 x^{12} - 50 x^{11} - 76 x^{10} - 135 x^{9} + 130 x^{8} + 29 x^{7} + 42 x^{6} - 76 x^{5} + 11 x^{4} - 8 x^{3} + 13 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-22378993818717495955548767565627=-\,3^{11}\cdot 1831^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.61$
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:  $3, 1831$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{47} a^{20} + \frac{12}{47} a^{19} + \frac{20}{47} a^{18} - \frac{14}{47} a^{17} - \frac{9}{47} a^{16} - \frac{12}{47} a^{15} - \frac{18}{47} a^{14} + \frac{3}{47} a^{13} - \frac{21}{47} a^{12} + \frac{3}{47} a^{11} + \frac{22}{47} a^{10} - \frac{1}{47} a^{9} + \frac{16}{47} a^{8} + \frac{3}{47} a^{7} - \frac{3}{47} a^{6} + \frac{10}{47} a^{5} + \frac{18}{47} a^{4} + \frac{2}{47} a^{3} - \frac{11}{47} a^{2} + \frac{10}{47} a + \frac{6}{47}$, $\frac{1}{905347652423} a^{21} - \frac{7009733765}{905347652423} a^{20} + \frac{216902865881}{905347652423} a^{19} + \frac{323405858872}{905347652423} a^{18} - \frac{323917225339}{905347652423} a^{17} + \frac{326039210861}{905347652423} a^{16} + \frac{15018399967}{905347652423} a^{15} + \frac{17789972227}{905347652423} a^{14} + \frac{630236487}{19262716009} a^{13} + \frac{356081718892}{905347652423} a^{12} - \frac{3437311634}{905347652423} a^{11} + \frac{1040380920}{905347652423} a^{10} - \frac{1931527219}{905347652423} a^{9} - \frac{263981466300}{905347652423} a^{8} - \frac{307455059123}{905347652423} a^{7} - \frac{234668290038}{905347652423} a^{6} + \frac{211410128850}{905347652423} a^{5} + \frac{426201096855}{905347652423} a^{4} - \frac{358627373030}{905347652423} a^{3} - \frac{200039368210}{905347652423} a^{2} + \frac{409680040182}{905347652423} a - \frac{408132685713}{905347652423}$

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:  \( -\frac{97902606020}{19262716009} a^{21} + \frac{121939211642}{19262716009} a^{20} - \frac{2182733406}{19262716009} a^{19} + \frac{966115408764}{19262716009} a^{18} - \frac{1023202462684}{19262716009} a^{17} - \frac{1005193668792}{19262716009} a^{16} - \frac{3692933865756}{19262716009} a^{15} + \frac{4608727529250}{19262716009} a^{14} + \frac{5411124004959}{19262716009} a^{13} + \frac{4493432812008}{19262716009} a^{12} - \frac{14850732943836}{19262716009} a^{11} - \frac{6149360661028}{19262716009} a^{10} + \frac{3440070117822}{19262716009} a^{9} + \frac{14980019208668}{19262716009} a^{8} - \frac{1702852996267}{19262716009} a^{7} - \frac{4087306024608}{19262716009} a^{6} - \frac{6382768745344}{19262716009} a^{5} + \frac{2342264256385}{19262716009} a^{4} + \frac{608678372872}{19262716009} a^{3} + \frac{1026929785830}{19262716009} a^{2} - \frac{350206773122}{19262716009} a + \frac{132316820371}{19262716009} \) (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:  \( 19742415.8494 \) (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

\(\Q(\sqrt{-3}) \), 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 $22$ R $22$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{11}$ $22$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
1831Data not computed