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: | $14$ |
| |
| Signature: | $(0, 7)$ |
| |
| Discriminant: |
\(-11856232199187\)
\(\medspace = -\,3^{7}\cdot 5421231001\)
|
| |
| Root discriminant: | \(8.59\) |
| |
| Galois root discriminant: | $3^{1/2}5421231001^{1/2}\approx 127529.18490682829$ | ||
| Ramified primes: |
\(3\), \(5421231001\)
|
| |
| Discriminant root field: | $\Q(\sqrt{-16263693003}$) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}) \) | ||
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: | $6$ |
| |
| 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 $6$)
|
| |
| Fundamental units: |
$a$, $a^{12}-a^{11}-2a^{7}+2a^{6}+a^{5}-a^{4}+a^{3}$, $a^{9}-a^{8}-2a^{4}+a^{3}+a^{2}-a+1$, $a^{12}-a^{11}-a^{7}+2a^{6}+a^{5}-a^{4}+a^{3}-a^{2}$, $a^{13}-a^{12}+a^{11}-a^{10}+a^{9}-2a^{8}+a^{7}-a^{6}+a^{5}-a^{3}+2a^{2}-a$, $a^{13}-a^{12}+a^{11}-2a^{8}+a^{7}-a^{6}-a^{5}+a^{4}+a^{2}$
|
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| Regulator: | \( 9.0466002324 \) |
|
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.7.2.7a1.2 | $x^{14} + 4 x^{9} + 2 x^{7} + 4 x^{4} + 4 x^{2} + 4$ | $2$ | $7$ | $7$ | $C_{14}$ | $$[\ ]_{2}^{7}$$ |
|
\(5421231001\)
| $\Q_{5421231001}$ | $x$ | $1$ | $1$ | $0$ | 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 $7$ | $1$ | $7$ | $0$ | $C_7$ | $$[\ ]^{7}$$ |