magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7112448, 0, 0, 0, 0, 35868, 0, 0, 0, 0, -420, 0, 0, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 420*x^10 + 35868*x^5 + 7112448)
gp: K = bnfinit(x^15 - 420*x^10 + 35868*x^5 + 7112448, 1)
Normalized defining polynomial
\( x^{15} - 420 x^{10} + 35868 x^{5} + 7112448 \)
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: | \(-654503601162655079739525888000000000000000=-\,2^{23}\cdot 3^{13}\cdot 5^{15}\cdot 7^{12}\cdot 41^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $613.38$ | 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: | $2, 3, 5, 7, 41$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{14} a^{5}$, $\frac{1}{14} a^{6}$, $\frac{1}{14} a^{7}$, $\frac{1}{28} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{28} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{77028} a^{10} + \frac{8}{917} a^{5} - \frac{1}{131}$, $\frac{1}{154056} a^{11} + \frac{4}{917} a^{6} - \frac{1}{262} a$, $\frac{1}{924336} a^{12} + \frac{139}{11004} a^{7} - \frac{263}{1572} a^{2}$, $\frac{1}{1848672} a^{13} + \frac{139}{22008} a^{8} + \frac{1309}{3144} a^{3}$, $\frac{1}{11092032} a^{14} + \frac{139}{132048} a^{9} - \frac{4979}{18864} a^{4}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{5}\times C_{5}$, which has order $25$ (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: | \( 177629767603198.8 \) (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
| 3.1.984.1, 5.1.9724050000.14 |
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 | R | R | R | ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ | ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ | ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$ |
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 | $x^{5} - 2$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ |
| 2.10.19.17 | $x^{10} - 2 x^{4} + 4 x^{2} - 10$ | $10$ | $1$ | $19$ | $F_{5}\times C_2$ | $[3]_{5}^{4}$ | |
| $3$ | 3.5.4.1 | $x^{5} - 3$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ |
| 3.10.9.2 | $x^{10} + 3$ | $10$ | $1$ | $9$ | $F_{5}\times C_2$ | $[\ ]_{10}^{4}$ | |
| $5$ | 5.15.15.40 | $x^{15} + 10 x^{14} + 5 x^{13} + 20 x^{12} + 20 x^{11} + 17 x^{10} + 10 x^{9} + 15 x^{8} + 12 x^{5} + 15 x^{3} + 10 x^{2} + 20 x + 17$ | $5$ | $3$ | $15$ | $F_5\times C_3$ | $[5/4]_{4}^{3}$ |
| $7$ | 7.5.4.1 | $x^{5} - 7$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ |
| 7.10.8.1 | $x^{10} - 7 x^{5} + 147$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| $41$ | $\Q_{41}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{41}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{41}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{41}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{41}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |