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Properties

Label 22.12.1781764468...3423.6
Degree $22$
Signature $[12, 5]$
Discriminant $-\,23^{20}\cdot 47^{3}$
Root discriminant $29.24$
Ramified primes $23, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T28

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -65, 705, -3122, 6866, -6693, -2411, 15029, -18181, 7283, 7597, -13923, 9671, -1948, -2650, 3076, -1667, 460, 26, -88, 42, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 10*x^21 + 42*x^20 - 88*x^19 + 26*x^18 + 460*x^17 - 1667*x^16 + 3076*x^15 - 2650*x^14 - 1948*x^13 + 9671*x^12 - 13923*x^11 + 7597*x^10 + 7283*x^9 - 18181*x^8 + 15029*x^7 - 2411*x^6 - 6693*x^5 + 6866*x^4 - 3122*x^3 + 705*x^2 - 65*x + 1)
 
gp: K = bnfinit(x^22 - 10*x^21 + 42*x^20 - 88*x^19 + 26*x^18 + 460*x^17 - 1667*x^16 + 3076*x^15 - 2650*x^14 - 1948*x^13 + 9671*x^12 - 13923*x^11 + 7597*x^10 + 7283*x^9 - 18181*x^8 + 15029*x^7 - 2411*x^6 - 6693*x^5 + 6866*x^4 - 3122*x^3 + 705*x^2 - 65*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - 10 x^{21} + 42 x^{20} - 88 x^{19} + 26 x^{18} + 460 x^{17} - 1667 x^{16} + 3076 x^{15} - 2650 x^{14} - 1948 x^{13} + 9671 x^{12} - 13923 x^{11} + 7597 x^{10} + 7283 x^{9} - 18181 x^{8} + 15029 x^{7} - 2411 x^{6} - 6693 x^{5} + 6866 x^{4} - 3122 x^{3} + 705 x^{2} - 65 x + 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-178176446876650757881387571453423=-\,23^{20}\cdot 47^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.24$
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:  $23, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{924207180721024804619} a^{21} - \frac{430113092941902278765}{924207180721024804619} a^{20} + \frac{39805276420178777666}{924207180721024804619} a^{19} + \frac{1096509094126161134}{3336488017043410847} a^{18} + \frac{210450717533563775796}{924207180721024804619} a^{17} + \frac{409804425824296184373}{924207180721024804619} a^{16} + \frac{59398931468727499186}{924207180721024804619} a^{15} + \frac{105033591465895272448}{924207180721024804619} a^{14} + \frac{286072638022791685140}{924207180721024804619} a^{13} + \frac{4459444447921199296}{924207180721024804619} a^{12} - \frac{382082488020992242483}{924207180721024804619} a^{11} + \frac{383106228904386282634}{924207180721024804619} a^{10} + \frac{304523984406824366237}{924207180721024804619} a^{9} + \frac{112228244840961530366}{924207180721024804619} a^{8} - \frac{117474817764336631226}{924207180721024804619} a^{7} + \frac{448484912860046135769}{924207180721024804619} a^{6} - \frac{311122115523088764560}{924207180721024804619} a^{5} + \frac{351381790637328911629}{924207180721024804619} a^{4} + \frac{368294505304384090745}{924207180721024804619} a^{3} - \frac{186701951637185652713}{924207180721024804619} a^{2} - \frac{426715091785121050728}{924207180721024804619} a + \frac{428982538929086872848}{924207180721024804619}$

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:  $16$
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:  \( 88859950.687 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T28:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 22528
The 208 conjugacy class representatives for t22n28 are not computed
Character table for t22n28 is not computed

Intermediate fields

\(\Q(\zeta_{23})^+\)

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

Sibling fields

Degree 22 siblings: 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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ R $22$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ $22$ $22$ R ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
23Data not computed
47Data not computed