magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -15, -81, -170, 16, 583, 621, -357, -864, -181, 341, 167, -27, -24, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 24*x^13 - 27*x^12 + 167*x^11 + 341*x^10 - 181*x^9 - 864*x^8 - 357*x^7 + 621*x^6 + 583*x^5 + 16*x^4 - 170*x^3 - 81*x^2 - 15*x - 1)
gp: K = bnfinit(x^15 - 24*x^13 - 27*x^12 + 167*x^11 + 341*x^10 - 181*x^9 - 864*x^8 - 357*x^7 + 621*x^6 + 583*x^5 + 16*x^4 - 170*x^3 - 81*x^2 - 15*x - 1, 1)
Normalized defining polynomial
\( x^{15} - 24 x^{13} - 27 x^{12} + 167 x^{11} + 341 x^{10} - 181 x^{9} - 864 x^{8} - 357 x^{7} + 621 x^{6} + 583 x^{5} + 16 x^{4} - 170 x^{3} - 81 x^{2} - 15 x - 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $15$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[15, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(23980112932460739050668032=2^{16}\cdot 3^{4}\cdot 37^{2}\cdot 53^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $49.20$ | 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: | $2, 3, 37, 53$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{111} a^{14} + \frac{10}{111} a^{13} - \frac{35}{111} a^{12} - \frac{44}{111} a^{11} - \frac{17}{37} a^{10} + \frac{53}{111} a^{9} + \frac{16}{111} a^{8} - \frac{38}{111} a^{7} + \frac{40}{111} a^{6} + \frac{22}{111} a^{5} + \frac{26}{111} a^{4} + \frac{18}{37} a^{3} + \frac{1}{3} a^{2} - \frac{44}{111} a - \frac{11}{111}$
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: | $14$ | 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: | \( 122000230.472 \) (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 4860 |
| The 63 conjugacy class representatives for [3^5]F(5)=3wrF(5) are not computed |
| Character table for [3^5]F(5)=3wrF(5) is not computed |
Intermediate fields
| 5.5.2382032.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 15 sibling: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 45 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 | R | ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ | $15$ | $15$ | ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ | ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ | R | ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ | $15$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ | R | $15$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.12.16.11 | $x^{12} + 20 x^{10} - 44 x^{8} - 4 x^{6} - 16 x^{4} - 48$ | $6$ | $2$ | $16$ | $C_3\times (C_3 : C_4)$ | $[2]_{3}^{6}$ | |
| $3$ | 3.3.4.3 | $x^{3} - 3 x^{2} + 12$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
| 3.12.0.1 | $x^{12} - x^{4} - x^{3} - x^{2} + x - 1$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| $37$ | 37.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 37.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.3.2.1 | $x^{3} - 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| $53$ | 53.3.0.1 | $x^{3} - x + 8$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 53.4.3.2 | $x^{4} - 212$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 53.4.3.2 | $x^{4} - 212$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 53.4.3.2 | $x^{4} - 212$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |