Normalized defining polynomial
\( x^{15} + 5 x^{13} - 10 x^{12} + 9 x^{11} - 36 x^{10} + 41 x^{9} - 30 x^{8} + 58 x^{7} - 24 x^{6} - 49 x^{5} + 74 x^{4} - 72 x^{3} + 80 x^{2} - 80 x + 32 \)
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: | \(4475148976045136896=2^{10}\cdot 13^{3}\cdot 347^{3}\cdot 47609\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $17.51$ | 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: | $2, 13, 347, 47609$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{8} a^{11} - \frac{3}{8} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{2} a^{6} + \frac{1}{8} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{11} + \frac{29}{64} a^{10} - \frac{1}{16} a^{9} + \frac{5}{64} a^{8} + \frac{5}{32} a^{7} + \frac{25}{64} a^{6} - \frac{1}{32} a^{4} + \frac{5}{16} a^{3} - \frac{1}{64} a^{2} + \frac{3}{16} a - \frac{5}{16}$, $\frac{1}{512} a^{13} - \frac{1}{128} a^{12} - \frac{31}{512} a^{11} - \frac{95}{256} a^{10} - \frac{179}{512} a^{9} - \frac{3}{8} a^{8} - \frac{123}{512} a^{7} + \frac{103}{256} a^{6} - \frac{1}{256} a^{5} + \frac{3}{64} a^{4} - \frac{41}{512} a^{3} + \frac{7}{256} a^{2} - \frac{11}{128} a + \frac{5}{64}$, $\frac{1}{77824} a^{14} - \frac{27}{38912} a^{13} + \frac{489}{77824} a^{12} + \frac{173}{4864} a^{11} - \frac{35159}{77824} a^{10} - \frac{16229}{38912} a^{9} + \frac{9029}{77824} a^{8} - \frac{299}{19456} a^{7} + \frac{6785}{38912} a^{6} + \frac{7711}{19456} a^{5} - \frac{26457}{77824} a^{4} + \frac{529}{4864} a^{3} + \frac{3963}{9728} a^{2} + \frac{383}{1216} a + \frac{1779}{4864}$
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: | \( 1965.65204624 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 933120 |
| The 108 conjugacy class representatives for [S(3)^5]S(5)=S(3)wrS(5) are not computed |
| Character table for [S(3)^5]S(5)=S(3)wrS(5) is not computed |
Intermediate fields
| 5.3.4511.1 |
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 | $15$ | $15$ | ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ | ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ | $15$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ | $15$ | ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 2.10.10.9 | $x^{10} - 15 x^{8} + 38 x^{6} - 18 x^{4} + 25 x^{2} - 63$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ | |
| $13$ | 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.3.0.1 | $x^{3} - 2 x + 6$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.6.0.1 | $x^{6} + x^{2} - 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 347 | Data not computed | ||||||
| 47609 | Data not computed | ||||||