Normalized defining polynomial
\( x^{14} - 4608 \)
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)
| |
Discriminant: | \(48376840350772245144010752\) \(\medspace = 2^{27}\cdot 3^{12}\cdot 7^{14}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(68.33\) | 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: | $2^{27/14}3^{6/7}7^{47/42}\approx 86.14376989441227$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{8}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{48}a^{7}$, $\frac{1}{96}a^{8}$, $\frac{1}{96}a^{9}$, $\frac{1}{192}a^{10}$, $\frac{1}{384}a^{11}$, $\frac{1}{384}a^{12}$, $\frac{1}{768}a^{13}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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: | $\frac{1}{48}a^{7}-1$, $\frac{1}{768}a^{13}+\frac{1}{384}a^{12}+\frac{1}{384}a^{11}+\frac{1}{192}a^{10}+\frac{1}{96}a^{9}+\frac{1}{96}a^{8}+\frac{1}{24}a^{7}+\frac{1}{8}a^{6}+\frac{1}{8}a^{5}+\frac{1}{4}a^{4}+\frac{1}{2}a^{3}+\frac{1}{2}a^{2}+a+3$, $\frac{1}{768}a^{13}-\frac{1}{384}a^{12}+\frac{1}{384}a^{11}-\frac{1}{192}a^{10}+\frac{1}{96}a^{9}-\frac{1}{96}a^{8}+\frac{1}{24}a^{7}-\frac{1}{8}a^{6}+\frac{1}{8}a^{5}-\frac{1}{4}a^{4}+\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+a-3$, $\frac{1}{96}a^{12}+\frac{3}{64}a^{10}+\frac{11}{96}a^{8}+\frac{1}{2}a^{6}+\frac{3}{2}a^{4}+5a^{2}+19$, $\frac{1}{64}a^{12}-\frac{13}{192}a^{10}+\frac{11}{48}a^{8}-\frac{5}{8}a^{6}+\frac{5}{4}a^{4}-\frac{1}{2}a^{2}-11$, $\frac{1}{96}a^{13}-\frac{11}{384}a^{12}+\frac{1}{32}a^{11}+\frac{13}{192}a^{10}-\frac{13}{48}a^{9}+\frac{49}{96}a^{8}-\frac{1}{4}a^{7}-a^{6}+\frac{7}{2}a^{5}-5a^{4}+2a^{3}+\frac{21}{2}a^{2}-23a+19$, $\frac{1103}{192}a^{13}-\frac{2279}{384}a^{12}-\frac{1073}{128}a^{11}+\frac{1855}{48}a^{10}-\frac{4871}{96}a^{9}-\frac{1931}{96}a^{8}+\frac{379}{2}a^{7}-\frac{1285}{4}a^{6}+\frac{599}{4}a^{5}+\frac{1091}{2}a^{4}-\frac{3519}{2}a^{3}+\frac{5641}{2}a^{2}-1010a-8851$ (assuming GRH) | 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: | \( 35271750.504034616 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 35271750.504034616 \cdot 1}{2\cdot\sqrt{48376840350772245144010752}}\cr\approx \mathstrut & 0.624047244318857 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times F_7$ (as 14T7):
A solvable group of order 84 |
The 14 conjugacy class representatives for $F_7 \times C_2$ |
Character table for $F_7 \times C_2$ |
Intermediate fields
\(\Q(\sqrt{2}) \), 7.1.38423222208.8 |
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 | R | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.14.0.1}{14} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{7}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.14.27.213 | $x^{14} + 4 x^{11} + 4 x^{9} + 4 x^{8} + 4 x^{7} + 4 x^{4} + 2$ | $14$ | $1$ | $27$ | $(C_7:C_3) \times C_2$ | $[3]_{7}^{3}$ |
\(3\) | 3.14.12.1 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
\(7\) | 7.7.7.5 | $x^{7} + 7 x + 7$ | $7$ | $1$ | $7$ | $F_7$ | $[7/6]_{6}$ |
7.7.7.5 | $x^{7} + 7 x + 7$ | $7$ | $1$ | $7$ | $F_7$ | $[7/6]_{6}$ |