Normalized defining polynomial
\( x^{14} - 2 x^{13} + 3 x^{12} - 4 x^{11} + 4 x^{10} - 4 x^{9} + 4 x^{8} - 5 x^{7} + 4 x^{6} - 4 x^{5} + \cdots + 1 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(52058057626129\) \(\medspace = 7215127^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(9.54\) | 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: | $7215127^{1/2}\approx 2686.098844048744$ | ||
Ramified primes: | \(7215127\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7}a^{13}-\frac{3}{7}a^{12}-\frac{1}{7}a^{11}-\frac{3}{7}a^{10}+\frac{3}{7}a^{8}+\frac{1}{7}a^{7}+\frac{1}{7}a^{6}+\frac{3}{7}a^{5}-\frac{3}{7}a^{3}-\frac{1}{7}a^{2}-\frac{3}{7}a+\frac{1}{7}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $a$, $a^{12}-a^{11}+a^{10}-2a^{9}+a^{8}-2a^{7}+2a^{6}-2a^{5}+a^{4}-a^{3}+2a^{2}-a+1$, $\frac{2}{7}a^{13}+\frac{1}{7}a^{12}+\frac{5}{7}a^{11}-\frac{6}{7}a^{10}-\frac{8}{7}a^{8}+\frac{2}{7}a^{7}-\frac{5}{7}a^{6}+\frac{6}{7}a^{5}-2a^{4}+\frac{1}{7}a^{3}-\frac{2}{7}a^{2}+\frac{8}{7}a-\frac{5}{7}$, $\frac{2}{7}a^{13}+\frac{1}{7}a^{12}-\frac{2}{7}a^{11}+\frac{1}{7}a^{10}-\frac{1}{7}a^{8}+\frac{2}{7}a^{7}-\frac{5}{7}a^{6}-\frac{8}{7}a^{5}+\frac{1}{7}a^{3}-\frac{2}{7}a^{2}-\frac{6}{7}a-\frac{5}{7}$, $\frac{1}{7}a^{13}-\frac{3}{7}a^{12}+\frac{6}{7}a^{11}-\frac{3}{7}a^{10}-\frac{4}{7}a^{8}+\frac{1}{7}a^{7}-\frac{6}{7}a^{6}+\frac{10}{7}a^{5}-\frac{3}{7}a^{3}-\frac{1}{7}a^{2}+\frac{4}{7}a+\frac{1}{7}$, $\frac{3}{7}a^{13}-\frac{2}{7}a^{12}+\frac{4}{7}a^{11}-\frac{2}{7}a^{10}+\frac{2}{7}a^{8}-\frac{4}{7}a^{7}-\frac{4}{7}a^{6}-\frac{5}{7}a^{5}-a^{4}-\frac{2}{7}a^{3}-\frac{3}{7}a^{2}-\frac{2}{7}a-\frac{4}{7}$, $\frac{2}{7}a^{13}+\frac{1}{7}a^{12}-\frac{2}{7}a^{11}+\frac{1}{7}a^{10}-\frac{1}{7}a^{8}+\frac{2}{7}a^{7}-\frac{5}{7}a^{6}-\frac{8}{7}a^{5}+\frac{1}{7}a^{3}-\frac{2}{7}a^{2}-\frac{6}{7}a+\frac{2}{7}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 9.48144292717 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 9.48144292717 \cdot 1}{2\cdot\sqrt{52058057626129}}\cr\approx \mathstrut & 0.161711036621 \end{aligned}\]
Galois group
$C_2^6.S_7$ (as 14T55):
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
7.5.7215127.1 |
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: | 14.8.1017906433737903368410527650477305199637303110503.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.7.0.1}{7} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.7.0.1}{7} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7215127\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |