Normalized defining polynomial
\( x^{14} - 3 x^{12} - 3 x^{11} + 6 x^{10} + 7 x^{9} - 5 x^{8} - 12 x^{7} + 2 x^{6} + 11 x^{5} + x^{4} + \cdots - 1 \)
Invariants
| Degree: | $14$ |
| |
| Signature: | $(2, 6)$ |
| |
| Discriminant: |
\(103886084489093\)
\(\medspace = 53\cdot 977^{2}\cdot 1433^{2}\)
|
| |
| Root discriminant: | \(10.03\) |
| |
| Galois root discriminant: | $53^{1/2}977^{1/2}1433^{1/2}\approx 8614.068318744634$ | ||
| Ramified primes: |
\(53\), \(977\), \(1433\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{53}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7}a^{13}-\frac{2}{7}a^{12}+\frac{1}{7}a^{11}+\frac{2}{7}a^{10}+\frac{2}{7}a^{9}+\frac{3}{7}a^{8}+\frac{3}{7}a^{7}+\frac{3}{7}a^{6}+\frac{3}{7}a^{5}-\frac{2}{7}a^{4}-\frac{2}{7}a^{3}-\frac{3}{7}a^{2}-\frac{3}{7}a+\frac{3}{7}$
| Monogenic: | Not computed | |
| 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: |
$a$, $\frac{10}{7}a^{13}+\frac{8}{7}a^{12}-\frac{25}{7}a^{11}-\frac{50}{7}a^{10}+\frac{27}{7}a^{9}+\frac{93}{7}a^{8}+\frac{9}{7}a^{7}-\frac{124}{7}a^{6}-\frac{54}{7}a^{5}+\frac{85}{7}a^{4}+\frac{64}{7}a^{3}-\frac{44}{7}a^{2}-\frac{44}{7}a+\frac{16}{7}$, $\frac{10}{7}a^{13}+\frac{1}{7}a^{12}-\frac{32}{7}a^{11}-\frac{36}{7}a^{10}+\frac{62}{7}a^{9}+\frac{93}{7}a^{8}-\frac{40}{7}a^{7}-\frac{152}{7}a^{6}-\frac{19}{7}a^{5}+\frac{127}{7}a^{4}+\frac{57}{7}a^{3}-\frac{65}{7}a^{2}-\frac{51}{7}a+\frac{23}{7}$, $\frac{2}{7}a^{13}+\frac{3}{7}a^{12}-\frac{5}{7}a^{11}-\frac{17}{7}a^{10}-\frac{3}{7}a^{9}+\frac{34}{7}a^{8}+\frac{27}{7}a^{7}-\frac{29}{7}a^{6}-\frac{50}{7}a^{5}+\frac{3}{7}a^{4}+\frac{38}{7}a^{3}+\frac{15}{7}a^{2}-\frac{13}{7}a-\frac{8}{7}$, $\frac{19}{7}a^{13}+\frac{11}{7}a^{12}-\frac{51}{7}a^{11}-\frac{88}{7}a^{10}+\frac{66}{7}a^{9}+\frac{176}{7}a^{8}+\frac{1}{7}a^{7}-\frac{237}{7}a^{6}-\frac{90}{7}a^{5}+\frac{172}{7}a^{4}+\frac{109}{7}a^{3}-\frac{85}{7}a^{2}-\frac{85}{7}a+\frac{43}{7}$, $\frac{22}{7}a^{13}+\frac{12}{7}a^{12}-\frac{55}{7}a^{11}-\frac{96}{7}a^{10}+\frac{65}{7}a^{9}+\frac{178}{7}a^{8}+\frac{10}{7}a^{7}-\frac{228}{7}a^{6}-\frac{95}{7}a^{5}+\frac{152}{7}a^{4}+\frac{103}{7}a^{3}-\frac{73}{7}a^{2}-\frac{73}{7}a+\frac{38}{7}$, $\frac{19}{7}a^{13}+\frac{11}{7}a^{12}-\frac{51}{7}a^{11}-\frac{88}{7}a^{10}+\frac{66}{7}a^{9}+\frac{176}{7}a^{8}+\frac{1}{7}a^{7}-\frac{237}{7}a^{6}-\frac{90}{7}a^{5}+\frac{172}{7}a^{4}+\frac{109}{7}a^{3}-\frac{85}{7}a^{2}-\frac{78}{7}a+\frac{43}{7}$
|
| |
| Regulator: | \( 14.647522546 \) |
| |
| Unit signature rank: | \( 2 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 14.647522546 \cdot 1}{2\cdot\sqrt{103886084489093}}\cr\approx \mathstrut & 0.17684577641 \end{aligned}\]
Galois group
$C_2^7.S_7$ (as 14T57):
| A non-solvable group of order 645120 |
| The 110 conjugacy class representatives for $C_2^7.S_7$ |
| Character table for $C_2^7.S_7$ |
Intermediate fields
| 7.3.1400041.1 |
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 siblings: | 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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.14.0.1}{14} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.14.0.1}{14} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ | R | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(53\)
| 53.1.2.1a1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 53.2.1.0a1.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 53.10.1.0a1.1 | $x^{10} + x^{6} + x^{4} + 27 x^{3} + 15 x^{2} + 29 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $$[\ ]^{10}$$ | |
|
\(977\)
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
|
\(1433\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $$[\ ]^{10}$$ |