Normalized defining polynomial
\( x^{15} - 5 x^{13} - 10 x^{12} - x^{11} - 4 x^{10} + 14 x^{9} + 108 x^{8} + 216 x^{7} + 144 x^{6} - 4 x^{5} - 40 x^{4} - 160 x^{3} - 320 x^{2} - 320 x - 128 \)
Invariants
| Degree: | $15$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[5, 5]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-309750028898725271546642432=-\,2^{14}\cdot 881^{6}\cdot 40433\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $58.35$ | 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: | $2, 881, 40433$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{16} a^{11} + \frac{3}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{16} a^{7} + \frac{1}{4} a^{6} - \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{192} a^{12} - \frac{1}{32} a^{11} - \frac{13}{192} a^{10} + \frac{1}{48} a^{9} + \frac{19}{192} a^{8} + \frac{17}{96} a^{7} - \frac{17}{96} a^{6} - \frac{1}{8} a^{5} - \frac{5}{12} a^{4} + \frac{1}{3} a^{3} + \frac{5}{16} a^{2} - \frac{1}{12} a - \frac{5}{12}$, $\frac{1}{1536} a^{13} - \frac{1}{384} a^{12} - \frac{25}{1536} a^{11} - \frac{11}{768} a^{10} + \frac{73}{512} a^{9} + \frac{7}{64} a^{8} - \frac{175}{768} a^{7} - \frac{47}{384} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{191}{384} a^{3} + \frac{61}{192} a^{2} + \frac{5}{96} a + \frac{7}{48}$, $\frac{1}{208896} a^{14} + \frac{7}{104448} a^{13} + \frac{191}{208896} a^{12} - \frac{755}{26112} a^{11} + \frac{6629}{69632} a^{10} - \frac{4371}{34816} a^{9} - \frac{18199}{104448} a^{8} - \frac{6019}{26112} a^{7} + \frac{2801}{26112} a^{6} + \frac{57}{136} a^{5} + \frac{19199}{52224} a^{4} - \frac{131}{6528} a^{3} + \frac{337}{6528} a^{2} + \frac{361}{1632} a - \frac{259}{1088}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 219428686.892 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 466560 |
| The 72 conjugacy class representatives for [S(3)^5]A(5)=S(3)wrA(5) are not computed |
| Character table for [S(3)^5]A(5)=S(3)wrA(5) is not computed |
Intermediate fields
| 5.5.3104644.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ | ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ | ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ | $15$ | $15$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ | $15$ | ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ | ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ | ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ | ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | $15$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.0.1 | $x^{2} - x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.3.2.1 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.4.4.2 | $x^{4} - x^{2} + 5$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
| 2.6.8.4 | $x^{6} + 2 x^{3} + 2 x^{2} + 2$ | $6$ | $1$ | $8$ | $S_4\times C_2$ | $[4/3, 4/3, 2]_{3}^{2}$ | |
| 881 | Data not computed | ||||||
| 40433 | Data not computed | ||||||