Normalized defining polynomial
\( x^{15} - 78 x^{13} - 137 x^{12} + 1971 x^{11} + 6309 x^{10} - 13289 x^{9} - 70686 x^{8} - 6660 x^{7} + 283696 x^{6} + 281520 x^{5} - 311052 x^{4} - 582802 x^{3} - 201060 x^{2} + 31908 x + 11116 \)
Invariants
| Degree: | $15$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[15, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(3292242610475668461259680831744=2^{8}\cdot 3^{20}\cdot 11^{6}\cdot 113^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $108.27$ | 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: | $2, 3, 11, 113$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{226} a^{12} - \frac{8}{113} a^{11} - \frac{29}{113} a^{10} - \frac{21}{226} a^{9} - \frac{93}{226} a^{8} - \frac{27}{226} a^{7} - \frac{103}{226} a^{6} - \frac{34}{113} a^{5} + \frac{3}{113} a^{4} - \frac{15}{113} a^{3} - \frac{2}{113} a^{2} + \frac{53}{113} a - \frac{8}{113}$, $\frac{1}{226} a^{13} - \frac{44}{113} a^{11} - \frac{45}{226} a^{10} + \frac{23}{226} a^{9} + \frac{67}{226} a^{8} - \frac{83}{226} a^{7} + \frac{46}{113} a^{6} + \frac{24}{113} a^{5} + \frac{33}{113} a^{4} - \frac{16}{113} a^{3} + \frac{21}{113} a^{2} + \frac{49}{113} a - \frac{15}{113}$, $\frac{1}{38898173661375923393822} a^{14} - \frac{16332926405912642792}{19449086830687961696911} a^{13} - \frac{23946530732984685121}{38898173661375923393822} a^{12} + \frac{9577299947668893462187}{38898173661375923393822} a^{11} + \frac{8073362930260791591043}{38898173661375923393822} a^{10} + \frac{9177519763461059548883}{19449086830687961696911} a^{9} - \frac{6683875684332996812580}{19449086830687961696911} a^{8} - \frac{3787448351856030291461}{38898173661375923393822} a^{7} - \frac{6995032228877127471563}{38898173661375923393822} a^{6} - \frac{8326629621482388913634}{19449086830687961696911} a^{5} + \frac{8351749945383590269600}{19449086830687961696911} a^{4} - \frac{9301193885134492852746}{19449086830687961696911} a^{3} + \frac{8702270933677347433121}{19449086830687961696911} a^{2} - \frac{168602095886405153731}{19449086830687961696911} a + \frac{725883396226401906300}{19449086830687961696911}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $14$ | 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: | \( 215812156738 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 233280 |
| The 48 conjugacy class representatives for [1/2.S(3)^5]A(5) |
| Character table for [1/2.S(3)^5]A(5) is not computed |
Intermediate fields
| 5.5.6180196.1 |
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 | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | R | $15$ | $15$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | $15$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | $15$ | ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ | $15$ |
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 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.4.0.1 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 2.6.6.8 | $x^{6} + 2 x + 2$ | $6$ | $1$ | $6$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
| 3 | Data not computed | ||||||
| $11$ | 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.3.2.1 | $x^{3} - 11$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 11.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 11.6.4.1 | $x^{6} + 220 x^{3} + 41503$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 113 | Data not computed | ||||||