magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -1, 21, 19, -181, -145, 833, 575, -2241, -1289, 3653, 1683, -3653, -1289, 2241, 575, -833, -145, 181, 19, -21, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 21*x^20 + 19*x^19 + 181*x^18 - 145*x^17 - 833*x^16 + 575*x^15 + 2241*x^14 - 1289*x^13 - 3653*x^12 + 1683*x^11 + 3653*x^10 - 1289*x^9 - 2241*x^8 + 575*x^7 + 833*x^6 - 145*x^5 - 181*x^4 + 19*x^3 + 21*x^2 - x - 1)
gp: K = bnfinit(x^22 - x^21 - 21*x^20 + 19*x^19 + 181*x^18 - 145*x^17 - 833*x^16 + 575*x^15 + 2241*x^14 - 1289*x^13 - 3653*x^12 + 1683*x^11 + 3653*x^10 - 1289*x^9 - 2241*x^8 + 575*x^7 + 833*x^6 - 145*x^5 - 181*x^4 + 19*x^3 + 21*x^2 - x - 1, 1)
Normalized defining polynomial
\( x^{22} - x^{21} - 21 x^{20} + 19 x^{19} + 181 x^{18} - 145 x^{17} - 833 x^{16} + 575 x^{15} + 2241 x^{14} - 1289 x^{13} - 3653 x^{12} + 1683 x^{11} + 3653 x^{10} - 1289 x^{9} - 2241 x^{8} + 575 x^{7} + 833 x^{6} - 145 x^{5} - 181 x^{4} + 19 x^{3} + 21 x^{2} - x - 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[22, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(386235767790620472019065537319289881=23^{20}\cdot 229\cdot 982789\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $41.46$ | 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: | $23, 229, 982789$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$
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: | \( 47528575548.1 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
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
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ | $22$ | R | ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{10}$ | ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ | $22$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $23$ | 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |
| 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ | |
| 229 | Data not computed | ||||||
| 982789 | Data not computed | ||||||