magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-50000, 0, 0, 0, 0, 3314, 0, 0, 0, 0, -92, 0, 0, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 92*x^10 + 3314*x^5 - 50000)
gp: K = bnfinit(x^15 - 92*x^10 + 3314*x^5 - 50000, 1)
Normalized defining polynomial
\( x^{15} - 92 x^{10} + 3314 x^{5} - 50000 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $15$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[1, 7]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-20096697792256000000000000000=-\,2^{23}\cdot 5^{15}\cdot 151^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $77.07$ | 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: | $2, 5, 151$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{1102} a^{10} + \frac{71}{551} a^{5} + \frac{88}{551}$, $\frac{1}{5510} a^{11} - \frac{1031}{2755} a^{6} - \frac{463}{2755} a$, $\frac{1}{55100} a^{12} - \frac{4648}{13775} a^{7} + \frac{10557}{27550} a^{2}$, $\frac{1}{275500} a^{13} - \frac{4648}{68875} a^{8} + \frac{65657}{137750} a^{3}$, $\frac{1}{2755000} a^{14} - \frac{211273}{688750} a^{9} + \frac{616657}{1377500} a^{4}$
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: | $7$ | 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: | \( 472801099.5191203 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$S_3\times F_5$ (as 15T11):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 120 |
| The 15 conjugacy class representatives for $F_5 \times S_3$ |
| Character table for $F_5 \times S_3$ |
Intermediate fields
| 3.1.1208.1, 5.1.50000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 30 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 | ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | $15$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.5.4.1 | $x^{5} - 2$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ |
| 2.10.19.1 | $x^{10} - 2$ | $10$ | $1$ | $19$ | $F_{5}\times C_2$ | $[3]_{5}^{4}$ | |
| $5$ | 5.5.5.2 | $x^{5} + 5 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $[5/4]_{4}$ |
| 5.10.10.7 | $x^{10} + 10 x^{8} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 12$ | $5$ | $2$ | $10$ | $F_{5}\times C_2$ | $[5/4]_{4}^{2}$ | |
| $151$ | $\Q_{151}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{151}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{151}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{151}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{151}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 151.2.1.2 | $x^{2} + 755$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.2.1.2 | $x^{2} + 755$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.2.1.2 | $x^{2} + 755$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.2.1.2 | $x^{2} + 755$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.2.1.2 | $x^{2} + 755$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |