↑

Properties

Label 22.0.61533497770...2816.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{36}\cdot 11^{23}$
Root discriminant $38.13$
Ramified primes $2, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T14

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1644, -5434, 6721, -44, 825, -4004, 7117, 4752, 2904, -814, 1309, 1840, 913, -44, 253, 308, -44, -66, 33, 0, -11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^20 + 33*x^18 - 66*x^17 - 44*x^16 + 308*x^15 + 253*x^14 - 44*x^13 + 913*x^12 + 1840*x^11 + 1309*x^10 - 814*x^9 + 2904*x^8 + 4752*x^7 + 7117*x^6 - 4004*x^5 + 825*x^4 - 44*x^3 + 6721*x^2 - 5434*x + 1644)
 
gp: K = bnfinit(x^22 - 11*x^20 + 33*x^18 - 66*x^17 - 44*x^16 + 308*x^15 + 253*x^14 - 44*x^13 + 913*x^12 + 1840*x^11 + 1309*x^10 - 814*x^9 + 2904*x^8 + 4752*x^7 + 7117*x^6 - 4004*x^5 + 825*x^4 - 44*x^3 + 6721*x^2 - 5434*x + 1644, 1)
 

Normalized defining polynomial

\( x^{22} - 11 x^{20} + 33 x^{18} - 66 x^{17} - 44 x^{16} + 308 x^{15} + 253 x^{14} - 44 x^{13} + 913 x^{12} + 1840 x^{11} + 1309 x^{10} - 814 x^{9} + 2904 x^{8} + 4752 x^{7} + 7117 x^{6} - 4004 x^{5} + 825 x^{4} - 44 x^{3} + 6721 x^{2} - 5434 x + 1644 \)

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:  \(-61533497770089105512253259111202816=-\,2^{36}\cdot 11^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.13$
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:  $2, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} + \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{9} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{18} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{8} a^{10} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{19} + \frac{1}{8} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{3}{8} a^{3}$, $\frac{1}{16} a^{20} - \frac{1}{16} a^{19} - \frac{1}{16} a^{18} - \frac{1}{8} a^{17} - \frac{3}{16} a^{12} + \frac{3}{16} a^{11} - \frac{1}{16} a^{10} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{16} a^{4} - \frac{1}{16} a^{3} - \frac{5}{16} a^{2} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{15666260320563965640003586638044528411984} a^{21} + \frac{356009219950138504864451829424145372821}{15666260320563965640003586638044528411984} a^{20} + \frac{13393055095823106651011312254626131083}{423412441096323395675772611839041308432} a^{19} + \frac{1277984110045833743164689303495169139}{26463277568520212229735788239940081777} a^{18} + \frac{60707043882536458053849144296995153124}{979141270035247852500224164877783025749} a^{17} - \frac{314874386932842288202790858285292153197}{3916565080140991410000896659511132102996} a^{16} - \frac{92667559491178089712358583546941274387}{3916565080140991410000896659511132102996} a^{15} + \frac{205340456772232546046865095184271333157}{1958282540070495705000448329755566051498} a^{14} - \frac{3669580245242028927987965184199660705691}{15666260320563965640003586638044528411984} a^{13} + \frac{16843650063496484209657045853485571457}{423412441096323395675772611839041308432} a^{12} - \frac{3846930805386272048704793680547449506221}{15666260320563965640003586638044528411984} a^{11} + \frac{403768037741106358204097304800092983829}{1958282540070495705000448329755566051498} a^{10} - \frac{483319081170787095720004205548346070043}{1958282540070495705000448329755566051498} a^{9} + \frac{210355490756620402153341151347971191511}{3916565080140991410000896659511132102996} a^{8} + \frac{277377160722565764118201996704270221543}{3916565080140991410000896659511132102996} a^{7} + \frac{1902181082965691735401071885382722688485}{3916565080140991410000896659511132102996} a^{6} - \frac{1594175347584752127119912533123506032411}{15666260320563965640003586638044528411984} a^{5} + \frac{669537852676628075667377927096838486245}{15666260320563965640003586638044528411984} a^{4} + \frac{1253727814837707580432686623005149396915}{15666260320563965640003586638044528411984} a^{3} + \frac{273270428081338551940812614603454676298}{979141270035247852500224164877783025749} a^{2} + \frac{484307073017912886557779586452641287076}{979141270035247852500224164877783025749} a + \frac{789657883572250429521396116891994416655}{1958282540070495705000448329755566051498}$

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:  \( -1 \) (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:  \( 984813482.893 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T14:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1320
The 13 conjugacy class representatives for t22n14
Character table for t22n14

Intermediate fields

\(\Q(\sqrt{-11}) \)

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

Sibling fields

Degree 12 sibling: data not computed
Degree 24 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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.16.8$x^{8} + 8 x^{5} + 12$$4$$2$$16$$S_4$$[8/3, 8/3]_{3}^{2}$
2.12.20.37$x^{12} - 6 x^{10} - x^{8} + 4 x^{6} + 3 x^{4} + 2 x^{2} - 7$$6$$2$$20$$S_4$$[8/3, 8/3]_{3}^{2}$
11Data not computed