Normalized defining polynomial
\( x^{14} - 2x^{13} - x^{11} + 2x^{10} + 2x^{8} + 2x^{6} + 2x^{4} - x^{3} - 2x + 1 \)
Invariants
| Degree: | $14$ |
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| Signature: | $[4, 5]$ |
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| Discriminant: |
\(-3907823013550959364\)
\(\medspace = -\,2^{2}\cdot 149^{4}\cdot 211^{4}\)
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| Root discriminant: | \(21.28\) |
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| Galois root discriminant: | $2\cdot 149^{2/3}211^{2/3}\approx 1992.2437733580762$ | ||
| Ramified primes: |
\(2\), \(149\), \(211\)
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| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{4}a^{13}+\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
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| Narrow class group: | $C_{2}$, which has order $2$ |
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Unit group
| Rank: | $8$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| |
| Fundamental units: |
$a$, $\frac{3}{2}a^{13}-\frac{5}{2}a^{12}-\frac{1}{2}a^{11}-2a^{10}+a^{9}+2a^{8}+3a^{7}+a^{6}+4a^{5}+2a^{4}+4a^{3}+\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{3}{2}$, $\frac{3}{4}a^{13}-\frac{5}{4}a^{12}+\frac{1}{4}a^{11}-2a^{10}+\frac{1}{2}a^{9}+\frac{1}{2}a^{8}+2a^{7}+2a^{6}+\frac{3}{2}a^{5}+\frac{3}{2}a^{4}+3a^{3}+\frac{5}{4}a^{2}+\frac{3}{4}a-\frac{5}{4}$, $\frac{1}{4}a^{13}+\frac{1}{4}a^{12}-\frac{5}{4}a^{11}-a^{10}+\frac{1}{2}a^{9}+\frac{1}{2}a^{8}+2a^{7}+\frac{5}{2}a^{5}-\frac{1}{2}a^{4}+3a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{7}{4}$, $a^{13}-a^{12}-2a^{11}-a^{10}+a^{9}+2a^{8}+2a^{7}+2a^{6}+2a^{5}+2a^{4}+2a^{3}-1$, $\frac{7}{4}a^{13}-\frac{9}{4}a^{12}-\frac{7}{4}a^{11}-3a^{10}+\frac{3}{2}a^{9}+\frac{5}{2}a^{8}+5a^{7}+2a^{6}+\frac{7}{2}a^{5}+\frac{7}{2}a^{4}+6a^{3}+\frac{9}{4}a^{2}-\frac{1}{4}a-\frac{13}{4}$, $\frac{3}{4}a^{13}-\frac{5}{4}a^{12}-\frac{3}{4}a^{11}+\frac{3}{2}a^{9}-\frac{1}{2}a^{8}+a^{6}+\frac{5}{2}a^{5}+\frac{1}{2}a^{4}+a^{3}+\frac{5}{4}a^{2}+\frac{7}{4}a-\frac{1}{4}$, $\frac{21}{4}a^{13}-\frac{31}{4}a^{12}-\frac{17}{4}a^{11}-7a^{10}+\frac{13}{2}a^{9}+\frac{7}{2}a^{8}+12a^{7}+7a^{6}+\frac{29}{2}a^{5}+\frac{15}{2}a^{4}+15a^{3}+\frac{11}{4}a^{2}+\frac{5}{4}a-\frac{39}{4}$
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| Regulator: | \( 13480.7995717 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{5}\cdot 13480.7995717 \cdot 1}{2\cdot\sqrt{3907823013550959364}}\cr\approx \mathstrut & 0.534241397160 \end{aligned}\]
Galois group
$C_2^7.A_7$ (as 14T56):
| A non-solvable group of order 322560 |
| The 64 conjugacy class representatives for $C_2^7.A_7$ |
| Character table for $C_2^7.A_7$ |
Intermediate fields
| 7.7.988410721.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| 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 | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.14.0.1}{14} }$ | ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.7.0.1}{7} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.14.0.1}{14} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.14.0.1}{14} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ |
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 2.10.1.0a1.1 | $x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $$[\ ]^{10}$$ | |
|
\(149\)
| $\Q_{149}$ | $x + 147$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{149}$ | $x + 147$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 149.2.1.0a1.1 | $x^{2} + 145 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 149.2.1.0a1.1 | $x^{2} + 145 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 149.2.1.0a1.1 | $x^{2} + 145 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 149.2.3.4a1.2 | $x^{6} + 435 x^{5} + 63081 x^{4} + 3050365 x^{3} + 126162 x^{2} + 1740 x + 157$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
|
\(211\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ | ||
| Deg $6$ | $3$ | $2$ | $4$ |