Normalized defining polynomial
\( x^{14} + 3 x^{12} - 10 x^{11} + 20 x^{10} + 10 x^{9} + 73 x^{8} - 6 x^{7} + 41 x^{6} + 234 x^{5} + 64 x^{4} + 186 x^{3} + 403 x^{2} + 156 x + 45 \)
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: | \(-56446250877184000000=-\,2^{14}\cdot 5^{6}\cdot 7^{6}\cdot 37^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $25.75$ | 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: | $2, 5, 7, 37$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{39632892543706692} a^{13} - \frac{4435222083126195}{13210964181235564} a^{12} - \frac{1296754549815385}{3302741045308891} a^{11} - \frac{9593988480710657}{19816446271853346} a^{10} + \frac{7871217682741963}{19816446271853346} a^{9} - \frac{4252267604803253}{9908223135926673} a^{8} + \frac{6755264923596589}{39632892543706692} a^{7} + \frac{6264918955227255}{13210964181235564} a^{6} + \frac{103279161970520}{9908223135926673} a^{5} + \frac{2564939420813425}{6605482090617782} a^{4} + \frac{452769663240971}{1042970856413334} a^{3} - \frac{1185334663684738}{3302741045308891} a^{2} + \frac{19208921641663363}{39632892543706692} a + \frac{2209652386799699}{13210964181235564}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $6$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{92449171}{10468062996} a^{13} + \frac{1254105}{3489354332} a^{12} - \frac{19285454}{872338583} a^{11} + \frac{443125439}{5234031498} a^{10} - \frac{866695111}{5234031498} a^{9} - \frac{384957535}{2617015749} a^{8} - \frac{5309188435}{10468062996} a^{7} + \frac{49285439}{3489354332} a^{6} + \frac{31084141}{2617015749} a^{5} - \frac{4379513649}{1744677166} a^{4} - \frac{86990627}{275475342} a^{3} - \frac{1060746897}{872338583} a^{2} - \frac{38012953225}{10468062996} a - \frac{2769353849}{3489354332} \) (order $4$) | 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: | \( 108233.425881 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 1764 |
| The 25 conjugacy class representatives for [1/2.F_42(7)^2]2 |
| Character table for [1/2.F_42(7)^2]2 is not computed |
Intermediate fields
| \(\Q(\sqrt{-1}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 21 siblings: | 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 | R | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ | R | R | ${\href{/LocalNumberField/11.14.0.1}{14} }$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.14.0.1}{14} }$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.14.0.1}{14} }$ | R | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.14.0.1}{14} }$ | ${\href{/LocalNumberField/47.14.0.1}{14} }$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.12.12.26 | $x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
| $5$ | $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.6.3.2 | $x^{6} - 25 x^{2} + 250$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| $7$ | 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.12.6.1 | $x^{12} + 294 x^{8} + 3430 x^{6} + 21609 x^{4} + 487403 x^{2} + 2941225$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
| $37$ | $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 37.6.4.3 | $x^{6} + 333 x^{3} + 34225$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 37.6.0.1 | $x^{6} - x + 20$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |