Normalized defining polynomial
\( x^{14} - 6x^{7} + 18 \)
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: | \(-157411105889382825984\) \(\medspace = -\,2^{20}\cdot 3^{12}\cdot 7^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(27.71\) | 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: | $2^{10/7}3^{6/7}7^{5/6}\approx 34.934319920652435$ | ||
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: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{3}a^{8}$, $\frac{1}{3}a^{9}$, $\frac{1}{3}a^{10}$, $\frac{1}{3}a^{11}$, $\frac{1}{21}a^{12}-\frac{1}{21}a^{11}-\frac{1}{7}a^{10}-\frac{2}{21}a^{8}+\frac{2}{21}a^{7}+\frac{2}{7}a^{6}-\frac{2}{7}a^{5}+\frac{1}{7}a^{4}+\frac{3}{7}a^{2}-\frac{3}{7}a-\frac{2}{7}$, $\frac{1}{21}a^{13}+\frac{1}{7}a^{11}-\frac{1}{7}a^{10}-\frac{2}{21}a^{9}+\frac{1}{21}a^{7}-\frac{1}{7}a^{5}+\frac{1}{7}a^{4}+\frac{3}{7}a^{3}+\frac{2}{7}a-\frac{2}{7}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{1}{3} a^{7} - 1 \) (order $4$) | 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: | $\frac{4}{21}a^{12}-\frac{4}{21}a^{11}+\frac{2}{21}a^{10}-\frac{1}{3}a^{9}+\frac{13}{21}a^{8}-\frac{13}{21}a^{7}+\frac{1}{7}a^{6}-\frac{8}{7}a^{5}+\frac{11}{7}a^{4}-a^{3}+\frac{5}{7}a^{2}-\frac{12}{7}a+\frac{13}{7}$, $\frac{4}{21}a^{12}-\frac{4}{21}a^{11}-\frac{5}{21}a^{10}+\frac{1}{3}a^{9}-\frac{1}{21}a^{8}-\frac{13}{21}a^{7}+\frac{1}{7}a^{6}-\frac{8}{7}a^{5}-\frac{3}{7}a^{4}+a^{3}-\frac{9}{7}a^{2}-\frac{12}{7}a+\frac{13}{7}$, $\frac{2}{21}a^{13}-\frac{1}{21}a^{12}-\frac{1}{7}a^{10}+\frac{10}{21}a^{9}-\frac{4}{7}a^{8}+\frac{1}{3}a^{7}-\frac{2}{7}a^{6}+\frac{1}{7}a^{4}+\frac{6}{7}a^{3}-\frac{10}{7}a^{2}+a+\frac{5}{7}$, $\frac{1}{7}a^{13}-\frac{2}{21}a^{12}+\frac{4}{21}a^{11}-\frac{1}{7}a^{10}-\frac{2}{7}a^{9}-\frac{1}{7}a^{8}-\frac{5}{7}a^{7}-\frac{4}{7}a^{6}+\frac{1}{7}a^{5}-\frac{6}{7}a^{4}+\frac{9}{7}a^{3}+\frac{1}{7}a^{2}+\frac{5}{7}a+\frac{5}{7}$, $\frac{1}{21}a^{13}+\frac{1}{21}a^{12}-\frac{5}{21}a^{11}-\frac{2}{7}a^{10}-\frac{3}{7}a^{9}+\frac{5}{21}a^{8}+\frac{10}{21}a^{7}+\frac{2}{7}a^{6}+\frac{4}{7}a^{5}+\frac{9}{7}a^{4}+\frac{3}{7}a^{3}+\frac{3}{7}a^{2}-\frac{15}{7}a-\frac{25}{7}$, $\frac{1}{7}a^{13}+\frac{1}{7}a^{12}+\frac{2}{7}a^{11}-\frac{4}{21}a^{10}+\frac{1}{21}a^{9}-\frac{13}{21}a^{8}+\frac{2}{21}a^{7}-\frac{8}{7}a^{6}-\frac{2}{7}a^{5}-\frac{8}{7}a^{4}+\frac{9}{7}a^{3}+\frac{2}{7}a^{2}+\frac{11}{7}a-\frac{5}{7}$ (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: | \( 106429.85466886447 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 106429.85466886447 \cdot 1}{4\cdot\sqrt{157411105889382825984}}\cr\approx \mathstrut & 0.819869765733122 \end{aligned}\] (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
\(\Q(\sqrt{-1}) \), 7.1.784147392.1 |
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 | R | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.14.20.8 | $x^{14} + 2 x^{13} + 2 x^{10} + 2 x^{7} + 2$ | $14$ | $1$ | $20$ | $(C_7:C_3) \times C_2$ | $[2]_{7}^{3}$ |
\(3\) | 3.14.12.1 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |