↑

Properties

Label 22.22.9843112975...2624.1
Degree $22$
Signature $[22, 0]$
Discriminant $2^{10}\cdot 11^{22}\cdot 17^{11}\cdot 3217^{5}$
Root discriminant $389.58$
Ramified primes $2, 11, 17, 3217$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T20

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 3168, 123904, -7920, -619520, 5544, 1208064, -1584, -1208064, 198, 704704, -9, -256256, 0, 59840, 0, -8976, 0, 836, 0, -44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 44*x^20 + 836*x^18 - 8976*x^16 + 59840*x^14 - 256256*x^12 - 9*x^11 + 704704*x^10 + 198*x^9 - 1208064*x^8 - 1584*x^7 + 1208064*x^6 + 5544*x^5 - 619520*x^4 - 7920*x^3 + 123904*x^2 + 3168*x + 16)
 
gp: K = bnfinit(x^22 - 44*x^20 + 836*x^18 - 8976*x^16 + 59840*x^14 - 256256*x^12 - 9*x^11 + 704704*x^10 + 198*x^9 - 1208064*x^8 - 1584*x^7 + 1208064*x^6 + 5544*x^5 - 619520*x^4 - 7920*x^3 + 123904*x^2 + 3168*x + 16, 1)
 

Normalized defining polynomial

\( x^{22} - 44 x^{20} + 836 x^{18} - 8976 x^{16} + 59840 x^{14} - 256256 x^{12} - 9 x^{11} + 704704 x^{10} + 198 x^{9} - 1208064 x^{8} - 1584 x^{7} + 1208064 x^{6} + 5544 x^{5} - 619520 x^{4} - 7920 x^{3} + 123904 x^{2} + 3168 x + 16 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(984311297528998947620977146874832596749227182758485402624=2^{10}\cdot 11^{22}\cdot 17^{11}\cdot 3217^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $389.58$
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:  $2, 11, 17, 3217$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{14} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3}$, $\frac{1}{6} a^{15} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a$, $\frac{1}{6} a^{16} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{12} a^{17} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{12} a^{18} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{12} a^{19} + \frac{1}{4} a^{8} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{24} a^{20} - \frac{3}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{24} a^{21} - \frac{3}{8} a^{10} - \frac{1}{4} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3}$

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:  $21$
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:  \( 13905869122000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T20:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4840
The 31 conjugacy class representatives for t22n20
Character table for t22n20 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \)

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

Sibling fields

Degree 22 sibling: data not computed
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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ $20{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ $20{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ $20{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ $20{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ $20{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.11.10.1$x^{11} - 2$$11$$1$$10$$F_{11}$$[\ ]_{11}^{10}$
11Data not computed
17Data not computed
3217Data not computed