Normalized defining polynomial
\( x^{14} - 3 x^{13} + 12 x^{12} - 13 x^{11} + 40 x^{10} - 31 x^{9} + 97 x^{8} - 37 x^{7} + 108 x^{6} - 23 x^{5} + 89 x^{4} - 4 x^{3} + 10 x^{2} + 1 \)
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: | \(-121781095161992523=-\,3^{7}\cdot 43^{2}\cdot 173539^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $16.61$ | 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: | $3, 43, 173539$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{27} a^{12} + \frac{1}{27} a^{11} - \frac{4}{27} a^{10} + \frac{5}{27} a^{9} + \frac{5}{27} a^{8} - \frac{1}{9} a^{7} + \frac{4}{9} a^{6} - \frac{10}{27} a^{5} - \frac{10}{27} a^{4} + \frac{2}{27} a^{3} + \frac{10}{27} a - \frac{4}{27}$, $\frac{1}{69818571} a^{13} + \frac{1133302}{69818571} a^{12} - \frac{437248}{69818571} a^{11} - \frac{4501663}{69818571} a^{10} - \frac{18525904}{69818571} a^{9} - \frac{1822871}{23272857} a^{8} + \frac{1498273}{23272857} a^{7} - \frac{24154354}{69818571} a^{6} + \frac{21217343}{69818571} a^{5} - \frac{9571267}{69818571} a^{4} - \frac{5697130}{23272857} a^{3} + \frac{18752068}{69818571} a^{2} - \frac{29673370}{69818571} a - \frac{6957034}{23272857}$
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: | \( -\frac{11778}{2585873} a^{13} + \frac{9213563}{69818571} a^{12} - \frac{28520071}{69818571} a^{11} + \frac{101884987}{69818571} a^{10} - \frac{115647047}{69818571} a^{9} + \frac{326255716}{69818571} a^{8} - \frac{90279578}{23272857} a^{7} + \frac{258511439}{23272857} a^{6} - \frac{304082489}{69818571} a^{5} + \frac{814967836}{69818571} a^{4} - \frac{182739407}{69818571} a^{3} + \frac{26000629}{2585873} a^{2} - \frac{16677412}{69818571} a + \frac{78868432}{69818571} \) (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: | \( 1964.35826515 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$S_7\times C_2$ (as 14T49):
| A non-solvable group of order 10080 |
| The 30 conjugacy class representatives for $S_7\times C_2$ |
| Character table for $S_7\times C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 7.7.67159593.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 14 sibling: | data not computed |
| Degree 28 sibling: | data not computed |
| Degree 42 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 | ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | ${\href{/LocalNumberField/29.14.0.1}{14} }$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/53.14.0.1}{14} }$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| $43$ | 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.5.0.1 | $x^{5} - x + 10$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 43.5.0.1 | $x^{5} - x + 10$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 173539 | Data not computed | ||||||