Normalized defining polynomial
\( x^{14} - 84 x^{12} + 2625 x^{10} - 38892 x^{8} - 926 x^{7} + 287042 x^{6} + 16646 x^{5} - 989016 x^{4} - 23548 x^{3} + 1195061 x^{2} - 341446 x - 146334 \)
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: | \(5443356357506393925712728014848=2^{14}\cdot 7^{21}\cdot 29^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $156.83$ | 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: | $2, 7, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{58} a^{10} + \frac{3}{58} a^{8} - \frac{7}{29} a^{6} - \frac{3}{58} a^{4} + \frac{1}{29} a^{3}$, $\frac{1}{174} a^{11} - \frac{1}{174} a^{10} - \frac{13}{87} a^{9} - \frac{1}{58} a^{8} - \frac{43}{174} a^{7} + \frac{7}{87} a^{6} + \frac{55}{174} a^{5} + \frac{5}{174} a^{4} - \frac{31}{174} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{5046} a^{12} + \frac{1}{1682} a^{10} + \frac{1}{6} a^{9} - \frac{239}{2523} a^{8} + \frac{1}{6} a^{7} - \frac{755}{1682} a^{6} + \frac{126}{841} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{1905192944396416734} a^{13} - \frac{43231370712113}{952596472198208367} a^{12} - \frac{3717209734270973}{1905192944396416734} a^{11} + \frac{3024002029321249}{635064314798805578} a^{10} - \frac{219649365623824934}{952596472198208367} a^{9} - \frac{273142033586607217}{1905192944396416734} a^{8} + \frac{886515692767527535}{1905192944396416734} a^{7} + \frac{52380288931424905}{952596472198208367} a^{6} + \frac{7302255700902484}{952596472198208367} a^{5} + \frac{20915412827829983}{65696308427462646} a^{4} + \frac{2806430597075617}{32848154213731323} a^{3} + \frac{163696289652134}{377564990962429} a^{2} - \frac{118599277097234}{377564990962429} a - \frac{67761363891367}{377564990962429}$
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: | \( 447182556479 \) (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
| \(\Q(\sqrt{7}) \) |
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 | ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | ${\href{/LocalNumberField/13.14.0.1}{14} }$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.14.0.1}{14} }$ | ${\href{/LocalNumberField/43.14.0.1}{14} }$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.6.6.3 | $x^{6} + 2 x^{4} + x^{2} - 7$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| 2.6.6.3 | $x^{6} + 2 x^{4} + x^{2} - 7$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| 7 | Data not computed | ||||||
| $29$ | 29.7.6.6 | $x^{7} - 464$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
| 29.7.0.1 | $x^{7} - x + 3$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |