Normalized defining polynomial
\( x^{14} - 4 x^{13} + 17 x^{12} - 46 x^{11} + 112 x^{10} - 216 x^{9} + 367 x^{8} - 520 x^{7} + 632 x^{6} - 649 x^{5} + 553 x^{4} - 381 x^{3} + 219 x^{2} - 99 x + 27 \)
Invariants
| Degree: | $14$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 7]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-446035689155158947=-\,3^{9}\cdot 13^{2}\cdot 366181^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $18.23$ | 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: | $3, 13, 366181$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{2}$, $\frac{1}{424521} a^{13} - \frac{62200}{424521} a^{12} - \frac{68656}{424521} a^{11} - \frac{128209}{424521} a^{10} - \frac{115388}{424521} a^{9} + \frac{48109}{141507} a^{8} - \frac{65180}{424521} a^{7} + \frac{183731}{424521} a^{6} - \frac{76366}{424521} a^{5} + \frac{118139}{424521} a^{4} - \frac{163223}{424521} a^{3} - \frac{19763}{47169} a^{2} + \frac{7804}{141507} a - \frac{2869}{47169}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $6$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{31520}{424521} a^{13} - \frac{106022}{424521} a^{12} + \frac{453952}{424521} a^{11} - \frac{1122830}{424521} a^{10} + \frac{2671787}{424521} a^{9} - \frac{1597078}{141507} a^{8} + \frac{7849418}{424521} a^{7} - \frac{10302866}{424521} a^{6} + \frac{11722831}{424521} a^{5} - \frac{10911464}{424521} a^{4} + \frac{8329952}{424521} a^{3} - \frac{503359}{47169} a^{2} + \frac{750449}{141507} a - \frac{55076}{47169} \) (order $6$) | 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: | \( 9313.85318657 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 56448 |
| The 27 conjugacy class representatives for [L(7)^2]2=L(7)wr2 |
| Character table for [L(7)^2]2=L(7)wr2 is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 16 sibling: | data not computed |
| Degree 28 siblings: | data not computed |
| Degree 42 sibling: | data not computed |
| Arithmetically equvalently 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 | ${\href{/LocalNumberField/2.14.0.1}{14} }$ | R | ${\href{/LocalNumberField/5.14.0.1}{14} }$ | ${\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ | ${\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | ${\href{/LocalNumberField/29.14.0.1}{14} }$ | ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ | ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.14.0.1}{14} }$ | ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ | ${\href{/LocalNumberField/59.14.0.1}{14} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.8.6.2 | $x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
| $13$ | $\Q_{13}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{13}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{13}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.7.0.1 | $x^{7} - 10 x + 2$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 366181 | Data not computed | ||||||