magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, -66, 286, 715, -2002, -3003, 6435, 6435, -11440, -8008, 12376, 6188, -8568, -3060, 3876, 969, -1140, -190, 210, 21, -22, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - x^22 - 22*x^21 + 21*x^20 + 210*x^19 - 190*x^18 - 1140*x^17 + 969*x^16 + 3876*x^15 - 3060*x^14 - 8568*x^13 + 6188*x^12 + 12376*x^11 - 8008*x^10 - 11440*x^9 + 6435*x^8 + 6435*x^7 - 3003*x^6 - 2002*x^5 + 715*x^4 + 286*x^3 - 66*x^2 - 12*x + 1)
gp: K = bnfinit(x^23 - x^22 - 22*x^21 + 21*x^20 + 210*x^19 - 190*x^18 - 1140*x^17 + 969*x^16 + 3876*x^15 - 3060*x^14 - 8568*x^13 + 6188*x^12 + 12376*x^11 - 8008*x^10 - 11440*x^9 + 6435*x^8 + 6435*x^7 - 3003*x^6 - 2002*x^5 + 715*x^4 + 286*x^3 - 66*x^2 - 12*x + 1, 1)
Normalized defining polynomial
\( x^{23} - x^{22} - 22 x^{21} + 21 x^{20} + 210 x^{19} - 190 x^{18} - 1140 x^{17} + 969 x^{16} + 3876 x^{15} - 3060 x^{14} - 8568 x^{13} + 6188 x^{12} + 12376 x^{11} - 8008 x^{10} - 11440 x^{9} + 6435 x^{8} + 6435 x^{7} - 3003 x^{6} - 2002 x^{5} + 715 x^{4} + 286 x^{3} - 66 x^{2} - 12 x + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $23$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[23, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(6111571184724799803076702357055363809=47^{22}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $39.76$ | 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: | $47$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(47\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{47}(1,·)$, $\chi_{47}(2,·)$, $\chi_{47}(3,·)$, $\chi_{47}(4,·)$, $\chi_{47}(6,·)$, $\chi_{47}(7,·)$, $\chi_{47}(8,·)$, $\chi_{47}(9,·)$, $\chi_{47}(12,·)$, $\chi_{47}(14,·)$, $\chi_{47}(16,·)$, $\chi_{47}(17,·)$, $\chi_{47}(18,·)$, $\chi_{47}(21,·)$, $\chi_{47}(24,·)$, $\chi_{47}(25,·)$, $\chi_{47}(27,·)$, $\chi_{47}(28,·)$, $\chi_{47}(32,·)$, $\chi_{47}(34,·)$, $\chi_{47}(36,·)$, $\chi_{47}(37,·)$, $\chi_{47}(42,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$
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: | $22$ | 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: | \( 68215743661.4 \) (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 cyclic group of order 23 |
| The 23 conjugacy class representatives for $C_{23}$ |
| Character table for $C_{23}$ is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $23$ | $23$ | $23$ | $23$ | $23$ | $23$ | $23$ | $23$ | $23$ | $23$ | $23$ | $23$ | $23$ | $23$ | R | $23$ | $23$ |
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 |
|---|---|---|---|---|---|---|---|
| 47 | Data not computed | ||||||