Normalized defining polynomial
\( x^{14} - x^{13} + x^{12} - x^{10} - x^{9} + x^{8} - x^{7} - x^{6} + 3x^{5} - 2x^{4} + x^{3} + x^{2} - x + 1 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-11856232199187\) \(\medspace = -\,3^{7}\cdot 5421231001\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(8.59\) | 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: | $3^{1/2}5421231001^{1/2}\approx 127529.18490682829$ | ||
Ramified primes: | \(3\), \(5421231001\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-16263693003}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -a^{13} + 2 a^{12} - a^{11} + a^{10} + a^{8} - 3 a^{7} + a^{6} - 2 a^{4} + 3 a^{3} - a^{2} + 1 \) (order $6$) | 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: | $a$, $a^{12}-a^{11}-2a^{7}+2a^{6}+a^{5}-a^{4}+a^{3}$, $a^{9}-a^{8}-2a^{4}+a^{3}+a^{2}-a+1$, $a^{12}-a^{11}-a^{7}+2a^{6}+a^{5}-a^{4}+a^{3}-a^{2}$, $a^{13}-a^{12}+a^{11}-a^{10}+a^{9}-2a^{8}+a^{7}-a^{6}+a^{5}-a^{3}+2a^{2}-a$, $a^{13}-a^{12}+a^{11}-2a^{8}+a^{7}-a^{6}-a^{5}+a^{4}+a^{2}$ | 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: | \( 9.0466002324 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 9.0466002324 \cdot 1}{6\cdot\sqrt{11856232199187}}\cr\approx \mathstrut & 0.16928554630 \end{aligned}\]
Galois group
$S_7\wr C_2$ (as 14T61):
A non-solvable group of order 50803200 |
The 135 conjugacy class representatives for $S_7\wr C_2$ |
Character table for $S_7\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{-3}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 28 siblings: | data not computed |
Degree 42 sibling: | 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 | ${\href{/padicField/2.14.0.1}{14} }$ | R | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.7.0.1}{7} }^{2}$ | ${\href{/padicField/17.14.0.1}{14} }$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.14.0.1}{14} }$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.14.7.2 | $x^{14} + 21 x^{12} + 189 x^{10} + 4 x^{9} + 945 x^{8} - 94 x^{7} + 2835 x^{6} - 630 x^{5} + 5107 x^{4} + 630 x^{3} + 5131 x^{2} + 1242 x + 2212$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ |
\(5421231001\) | $\Q_{5421231001}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{5421231001}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |