Normalized defining polynomial
\( x^{14} - x^{13} + 3187 x^{12} - 17747 x^{11} - 109151356 x^{10} + 92024377 x^{9} + 214767646363 x^{8} + \cdots + 86\!\cdots\!07 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-792073307198230192780090690967483501665367\) \(\medspace = -\,23^{5}\cdot 239^{7}\cdot 431^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(983.49\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $23^{1/2}239^{1/2}431^{1/2}\approx 1539.2228558594106$ | ||
Ramified primes: | \(23\), \(239\), \(431\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-2369207}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{23}a^{11}-\frac{10}{23}a^{10}+\frac{7}{23}a^{9}-\frac{2}{23}a^{8}+\frac{5}{23}a^{7}-\frac{2}{23}a^{6}-\frac{5}{23}a^{5}-\frac{5}{23}a^{3}-\frac{9}{23}a^{2}-\frac{9}{23}a-\frac{9}{23}$, $\frac{1}{23}a^{12}-\frac{1}{23}a^{10}-\frac{1}{23}a^{9}+\frac{8}{23}a^{8}+\frac{2}{23}a^{7}-\frac{2}{23}a^{6}-\frac{4}{23}a^{5}-\frac{5}{23}a^{4}+\frac{10}{23}a^{3}-\frac{7}{23}a^{2}-\frac{7}{23}a+\frac{2}{23}$, $\frac{1}{10\!\cdots\!43}a^{13}-\frac{99\!\cdots\!54}{10\!\cdots\!43}a^{12}+\frac{15\!\cdots\!69}{10\!\cdots\!43}a^{11}-\frac{17\!\cdots\!03}{10\!\cdots\!43}a^{10}-\frac{48\!\cdots\!09}{10\!\cdots\!43}a^{9}-\frac{38\!\cdots\!73}{10\!\cdots\!43}a^{8}-\frac{50\!\cdots\!88}{10\!\cdots\!43}a^{7}-\frac{29\!\cdots\!13}{10\!\cdots\!43}a^{6}+\frac{59\!\cdots\!66}{10\!\cdots\!43}a^{5}+\frac{16\!\cdots\!79}{10\!\cdots\!43}a^{4}+\frac{30\!\cdots\!66}{10\!\cdots\!43}a^{3}-\frac{35\!\cdots\!54}{10\!\cdots\!43}a^{2}-\frac{30\!\cdots\!45}{10\!\cdots\!43}a-\frac{37\!\cdots\!97}{10\!\cdots\!43}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $
Galois group
$C_2^6.S_7$ (as 14T54):
A non-solvable group of order 322560 |
The 55 conjugacy class representatives for $C_2^6.S_7$ |
Character table for $C_2^6.S_7$ |
Intermediate fields
7.5.2369207.1 |
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 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.7.0.1}{7} }^{2}$ | ${\href{/padicField/17.7.0.1}{7} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(23\) | 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.6.3.1 | $x^{6} + 1058 x^{2} - 219006$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(239\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $8$ | $2$ | $4$ | $4$ | ||||
\(431\) | Deg $4$ | $2$ | $2$ | $2$ | |||
Deg $10$ | $2$ | $5$ | $5$ |