Normalized defining polynomial
\( x^{14} - x^{12} - 2x^{9} + 3x^{7} - 2x^{5} - x^{2} + 1 \)
Invariants
| Degree: | $14$ |
| |
| Signature: | $(2, 6)$ |
| |
| Discriminant: |
\(94241555580481\)
\(\medspace = 43^{2}\cdot 401^{2}\cdot 563^{2}\)
|
| |
| Root discriminant: | \(9.96\) |
| |
| Galois root discriminant: | $43^{1/2}401^{1/2}563^{1/2}\approx 3115.735707661996$ | ||
| Ramified primes: |
\(43\), \(401\), \(563\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
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
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $7$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a$, $a^{13}-a^{12}-2a^{11}+2a^{10}-a^{9}-2a^{8}+4a^{7}+3a^{6}-3a^{5}-a^{4}+2a^{3}-2a^{2}+2$, $a^{12}-a^{11}+a^{10}-2a^{8}+a^{6}-a^{5}+2a^{3}-a^{2}-2a+1$, $a^{13}-a^{9}-a^{8}-a^{7}+a^{6}+a^{5}-a^{3}-a^{2}-1$, $a^{13}-a^{12}+a^{10}-a^{9}-a^{8}+a^{7}+a^{6}-2a^{5}+a^{3}-1$, $a^{13}-a^{9}-2a^{8}+a^{6}+a^{4}-a^{2}-a$, $a^{12}-a^{11}+a^{9}-a^{8}-a^{7}+a^{6}-2a^{4}+a^{2}+1$
|
| |
| Regulator: | \( 13.4715273772 \) |
| |
| Unit signature rank: | \( 2 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 13.4715273772 \cdot 1}{2\cdot\sqrt{94241555580481}}\cr\approx \mathstrut & 0.170767342837 \end{aligned}\]
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
| 7.3.9707809.1 |
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: | 14.2.8125471312815539309958298932405426404108029721569.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.7.0.1}{7} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(43\)
| 43.2.1.0a1.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 43.2.2.2a1.2 | $x^{4} + 84 x^{3} + 1770 x^{2} + 252 x + 52$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 43.8.1.0a1.1 | $x^{8} + x^{4} + 39 x^{3} + 20 x^{2} + 24 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $$[\ ]^{8}$$ | |
|
\(401\)
| Deg $4$ | $2$ | $2$ | $2$ | |||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $$[\ ]^{5}$$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $$[\ ]^{5}$$ | ||
|
\(563\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |