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Properties

Label 22.22.1747188397...3553.1
Degree $22$
Signature $[22, 0]$
Discriminant $1297^{11}$
Root discriminant $36.01$
Ramified prime $1297$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{11}$ (as 22T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1297, 0, 35007, 0, -220847, 0, 569938, 0, -702109, 0, 481348, 0, -199186, 0, 51512, 0, -8336, 0, 816, 0, -44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 44*x^20 + 816*x^18 - 8336*x^16 + 51512*x^14 - 199186*x^12 + 481348*x^10 - 702109*x^8 + 569938*x^6 - 220847*x^4 + 35007*x^2 - 1297)
 
gp: K = bnfinit(x^22 - 44*x^20 + 816*x^18 - 8336*x^16 + 51512*x^14 - 199186*x^12 + 481348*x^10 - 702109*x^8 + 569938*x^6 - 220847*x^4 + 35007*x^2 - 1297, 1)
 

Normalized defining polynomial

\( x^{22} - 44 x^{20} + 816 x^{18} - 8336 x^{16} + 51512 x^{14} - 199186 x^{12} + 481348 x^{10} - 702109 x^{8} + 569938 x^{6} - 220847 x^{4} + 35007 x^{2} - 1297 \)

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:  \(17471883970840462300304775614373553=1297^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.01$
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:  $1297$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{10} a^{16} + \frac{1}{5} a^{14} + \frac{1}{10} a^{12} + \frac{2}{5} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{10} a^{6} + \frac{2}{5} a^{4} - \frac{1}{2} a^{3} - \frac{3}{10} a^{2} - \frac{1}{2} a + \frac{2}{5}$, $\frac{1}{10} a^{17} + \frac{1}{5} a^{15} + \frac{1}{10} a^{13} - \frac{1}{10} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{10} a^{7} + \frac{2}{5} a^{5} - \frac{3}{10} a^{3} - \frac{1}{10} a - \frac{1}{2}$, $\frac{1}{2050} a^{18} - \frac{1}{82} a^{16} + \frac{7}{2050} a^{14} + \frac{147}{2050} a^{12} - \frac{154}{1025} a^{10} - \frac{961}{2050} a^{8} - \frac{1}{2} a^{7} + \frac{161}{2050} a^{6} - \frac{1}{2} a^{5} - \frac{438}{1025} a^{4} - \frac{68}{205} a^{2} - \frac{633}{2050}$, $\frac{1}{2050} a^{19} - \frac{1}{82} a^{17} + \frac{7}{2050} a^{15} + \frac{147}{2050} a^{13} - \frac{154}{1025} a^{11} - \frac{961}{2050} a^{9} - \frac{1}{2} a^{8} + \frac{161}{2050} a^{7} - \frac{1}{2} a^{6} - \frac{438}{1025} a^{5} - \frac{68}{205} a^{3} - \frac{633}{2050} a$, $\frac{1}{18511120616750} a^{20} + \frac{2927096123}{18511120616750} a^{18} - \frac{652472479893}{18511120616750} a^{16} + \frac{1203748839829}{9255560308375} a^{14} - \frac{1127631712051}{9255560308375} a^{12} - \frac{1256632259119}{3702224123350} a^{10} - \frac{1}{2} a^{9} + \frac{538652610483}{18511120616750} a^{8} - \frac{3268362941224}{9255560308375} a^{6} - \frac{1}{2} a^{5} + \frac{397907936247}{18511120616750} a^{4} - \frac{1}{2} a^{3} + \frac{9082411779777}{18511120616750} a^{2} - \frac{1295366620242}{9255560308375}$, $\frac{1}{18511120616750} a^{21} + \frac{2927096123}{18511120616750} a^{19} - \frac{652472479893}{18511120616750} a^{17} + \frac{1203748839829}{9255560308375} a^{15} - \frac{1127631712051}{9255560308375} a^{13} + \frac{297239901278}{1851112061675} a^{11} - \frac{1}{2} a^{10} + \frac{538652610483}{18511120616750} a^{9} - \frac{3268362941224}{9255560308375} a^{7} - \frac{1}{2} a^{6} + \frac{397907936247}{18511120616750} a^{5} + \frac{9082411779777}{18511120616750} a^{3} - \frac{1}{2} a^{2} + \frac{6664827067891}{18511120616750} a - \frac{1}{2}$

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:  \( 10757085477.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{11}$ (as 22T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 22
The 7 conjugacy class representatives for $D_{11}$
Character table for $D_{11}$

Intermediate fields

\(\Q(\sqrt{1297}) \), 11.11.3670285774226257.1 x11

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

Sibling fields

Degree 11 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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{11}$

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
1297Data not computed