Normalized defining polynomial
\( x^{15} - 6 x^{13} - 4 x^{12} + 9 x^{11} + 12 x^{10} - 77 x^{9} - 162 x^{8} - 27 x^{7} + 192 x^{6} - 27 x^{5} - 714 x^{4} - 1064 x^{3} - 720 x^{2} - 240 x - 32 \)
Invariants
| Degree: | $15$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[3, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(16295544285146337272832=2^{10}\cdot 3^{16}\cdot 7177^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $30.26$ | 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: | $2, 3, 7177$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{8} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{8} a^{7} - \frac{1}{2} a^{6} + \frac{3}{8} a^{5} - \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{11} + \frac{9}{32} a^{10} + \frac{3}{8} a^{9} - \frac{15}{64} a^{8} + \frac{13}{32} a^{7} + \frac{19}{64} a^{6} - \frac{3}{8} a^{5} - \frac{31}{64} a^{4} + \frac{11}{32} a^{3} + \frac{13}{64} a^{2} - \frac{1}{16} a + \frac{5}{16}$, $\frac{1}{512} a^{13} - \frac{1}{128} a^{12} + \frac{11}{256} a^{11} - \frac{35}{128} a^{10} - \frac{191}{512} a^{9} + \frac{15}{64} a^{8} + \frac{223}{512} a^{7} - \frac{63}{256} a^{6} + \frac{81}{512} a^{5} - \frac{27}{128} a^{4} - \frac{159}{512} a^{3} + \frac{81}{256} a^{2} - \frac{9}{128} a - \frac{21}{64}$, $\frac{1}{4096} a^{14} + \frac{1}{2048} a^{13} - \frac{1}{2048} a^{12} - \frac{1}{512} a^{11} - \frac{7}{4096} a^{10} - \frac{1}{2048} a^{9} - \frac{81}{4096} a^{8} - \frac{81}{1024} a^{7} - \frac{675}{4096} a^{6} - \frac{579}{2048} a^{5} + \frac{1753}{4096} a^{4} - \frac{163}{512} a^{3} + \frac{53}{512} a^{2} + \frac{1}{32} a + \frac{1}{256}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $8$ | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 212744.042378 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 466560 |
| The 72 conjugacy class representatives for [1/2.S(3)^5]S(5) are not computed |
| Character table for [1/2.S(3)^5]S(5) is not computed |
Intermediate fields
| 5.1.7177.1 |
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 | R | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ | $15$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ | ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | $15$ | $15$ | $15$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ | ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 2.10.10.2 | $x^{10} - 5 x^{8} + 10 x^{6} - 2 x^{4} - 11 x^{2} + 39$ | $2$ | $5$ | $10$ | $C_2^4 : C_5$ | $[2, 2, 2, 2]^{5}$ | |
| $3$ | 3.3.4.3 | $x^{3} - 3 x^{2} + 12$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
| 3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ | |
| 3.9.9.2 | $x^{9} + 18 x^{3} + 27 x + 27$ | $3$ | $3$ | $9$ | $C_3^2 : S_3 $ | $[3/2, 3/2]_{2}^{3}$ | |
| 7177 | Data not computed | ||||||