Normalized defining polynomial
\( x^{15} - 17028 x^{10} + 265060026 x^{5} - 41497747632 \)
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: | \(-898400045220655932623997317447073792000000000=-\,2^{23}\cdot 3^{13}\cdot 5^{9}\cdot 11^{13}\cdot 251^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $992.88$ | 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: | $2, 3, 5, 11, 251$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{3} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{9} a^{6}$, $\frac{1}{27} a^{7} + \frac{1}{3} a^{2}$, $\frac{1}{81} a^{8} + \frac{1}{9} a^{3}$, $\frac{1}{243} a^{9} - \frac{2}{27} a^{4}$, $\frac{1}{3100963338} a^{10} - \frac{565916}{15661431} a^{5} - \frac{40374}{193351}$, $\frac{1}{9302890014} a^{11} - \frac{2306075}{46984293} a^{6} - \frac{13458}{193351} a$, $\frac{1}{3069953704620} a^{12} - \frac{1}{23257225035} a^{11} + \frac{1}{15504816690} a^{10} + \frac{1}{1215} a^{9} - \frac{2}{405} a^{8} + \frac{79764356}{7752408345} a^{7} - \frac{11049281}{234921465} a^{6} - \frac{4046234}{78307155} a^{5} + \frac{16}{135} a^{4} + \frac{1}{45} a^{3} + \frac{2046829}{12761166} a^{2} + \frac{220267}{966755} a - \frac{427076}{966755}$, $\frac{1}{9209861113860} a^{13} - \frac{1}{23257225035} a^{11} + \frac{1}{15504816690} a^{10} + \frac{2}{1215} a^{9} - \frac{18502277}{4651445007} a^{8} - \frac{1}{135} a^{7} + \frac{9832627}{234921465} a^{6} - \frac{461215}{15661431} a^{5} + \frac{1}{27} a^{4} - \frac{15288187}{191417490} a^{3} + \frac{2}{15} a^{2} + \frac{26916}{966755} a - \frac{40374}{966755}$, $\frac{1}{607850833514760} a^{14} - \frac{1}{46514450070} a^{11} - \frac{1}{15504816690} a^{10} - \frac{2159820277}{1534976852310} a^{9} - \frac{2}{405} a^{8} - \frac{1}{135} a^{7} - \frac{2914402}{234921465} a^{6} - \frac{1174243}{78307155} a^{5} + \frac{980082761}{12633554340} a^{4} - \frac{2}{45} a^{3} - \frac{7}{15} a^{2} + \frac{80032}{193351} a - \frac{152977}{966755}$
Class group and class number
$C_{5}\times C_{85}$, which has order $425$ (assuming GRH)
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: | \( 606628696626962.5 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$S_3\times F_5$ (as 15T11):
| 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
| 3.1.66264.1, 5.1.2371842000.3 |
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 | ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ | $15$ | ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ | $15$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
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 | $x^{5} - 3$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ |
| 3.10.9.1 | $x^{10} - 3$ | $10$ | $1$ | $9$ | $F_{5}\times C_2$ | $[\ ]_{10}^{4}$ | |
| $5$ | 5.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 5.12.9.2 | $x^{12} - 10 x^{8} + 25 x^{4} - 500$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |
| $11$ | 11.5.4.3 | $x^{5} + 33$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.10.9.5 | $x^{10} - 8019$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
| 251 | Data not computed | ||||||