Normalized defining polynomial
\( x^{14} - 6 x^{13} - 59 x^{12} + 380 x^{11} + 631 x^{10} - 6114 x^{9} + 6477 x^{8} + 12096 x^{7} - 21795 x^{6} - 1290 x^{5} + 16631 x^{4} - 6556 x^{3} - 835 x^{2} + 426 x + 47 \)
Invariants
| Degree: | $14$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[14, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(22128144096926782426357811904512=2^{27}\cdot 7^{14}\cdot 79^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $173.35$ | 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: | $2, 7, 79$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{7} a^{4} - \frac{1}{7} a^{3} + \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{5} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{6} + \frac{1}{7} a^{3} - \frac{2}{7} a^{2} - \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{49} a^{7} + \frac{1}{49} a^{6} - \frac{3}{49} a^{5} + \frac{2}{49} a^{4} + \frac{22}{49} a^{3} - \frac{17}{49} a^{2} - \frac{19}{49} a - \frac{17}{49}$, $\frac{1}{49} a^{8} + \frac{3}{49} a^{6} - \frac{2}{49} a^{5} - \frac{1}{49} a^{4} - \frac{4}{49} a^{3} + \frac{19}{49} a^{2} + \frac{9}{49} a + \frac{3}{49}$, $\frac{1}{49} a^{9} + \frac{2}{49} a^{6} + \frac{1}{49} a^{5} - \frac{3}{49} a^{4} + \frac{9}{49} a^{3} - \frac{17}{49} a^{2} - \frac{24}{49} a + \frac{23}{49}$, $\frac{1}{343} a^{10} + \frac{3}{343} a^{9} - \frac{2}{343} a^{8} + \frac{1}{343} a^{7} - \frac{2}{49} a^{6} + \frac{1}{49} a^{5} - \frac{2}{49} a^{4} - \frac{53}{343} a^{3} - \frac{117}{343} a^{2} + \frac{148}{343} a - \frac{53}{343}$, $\frac{1}{343} a^{11} + \frac{3}{343} a^{9} - \frac{3}{343} a^{7} + \frac{3}{49} a^{6} - \frac{18}{343} a^{4} + \frac{9}{49} a^{3} + \frac{86}{343} a^{2} - \frac{19}{49} a - \frac{121}{343}$, $\frac{1}{2401} a^{12} + \frac{2}{2401} a^{11} - \frac{3}{2401} a^{10} - \frac{5}{2401} a^{9} - \frac{19}{2401} a^{8} + \frac{2}{2401} a^{7} - \frac{3}{49} a^{6} + \frac{171}{2401} a^{5} + \frac{104}{2401} a^{4} - \frac{135}{2401} a^{3} + \frac{209}{2401} a^{2} + \frac{594}{2401} a - \frac{904}{2401}$, $\frac{1}{2401} a^{13} + \frac{1}{2401} a^{10} + \frac{12}{2401} a^{9} - \frac{9}{2401} a^{8} + \frac{24}{2401} a^{7} - \frac{25}{2401} a^{6} - \frac{6}{343} a^{5} - \frac{4}{343} a^{4} - \frac{746}{2401} a^{3} - \frac{741}{2401} a^{2} + \frac{15}{2401} a - \frac{117}{2401}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 493281649946 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_7^2:C_6$ (as 14T14):
| A solvable group of order 294 |
| The 17 conjugacy class representatives for $C_7^2:C_6$ |
| Character table for $C_7^2:C_6$ |
Intermediate fields
| \(\Q(\sqrt{2}) \) |
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 | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | ${\href{/LocalNumberField/13.14.0.1}{14} }$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.14.0.1}{14} }$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ | ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ | ${\href{/LocalNumberField/43.14.0.1}{14} }$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.14.27.213 | $x^{14} + 2 x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{8} + 4 x^{6} + 4 x^{5} + 4 x^{2} - 2$ | $14$ | $1$ | $27$ | $(C_7:C_3) \times C_2$ | $[3]_{7}^{3}$ |
| $7$ | $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 7.3.2.2 | $x^{3} - 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.3.2.2 | $x^{3} - 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.7.10.3 | $x^{7} + 14 x^{4} + 7$ | $7$ | $1$ | $10$ | $C_7:C_3$ | $[5/3]_{3}$ | |
| 79 | Data not computed | ||||||