Normalized defining polynomial
\( x^{8} - 20x^{6} + 126x^{4} - 312x^{2} + 468 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(126142572920832\)
\(\medspace = 2^{22}\cdot 3^{4}\cdot 13^{5}\)
|
| |
| Root discriminant: | \(57.89\) |
| |
| Galois root discriminant: | $2^{11/4}3^{3/4}13^{3/4}\approx 104.98589220070072$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{114}a^{6}+\frac{13}{114}a^{4}+\frac{7}{19}a^{2}+\frac{8}{19}$, $\frac{1}{114}a^{7}+\frac{13}{114}a^{5}+\frac{7}{19}a^{3}+\frac{8}{19}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{2}\times C_{4}$, which has order $8$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{57}a^{6}-\frac{31}{114}a^{4}+\frac{14}{19}a^{2}-\frac{22}{19}$, $\frac{3}{38}a^{6}-\frac{37}{38}a^{4}+\frac{44}{19}a^{2}-\frac{118}{19}$, $\frac{2}{19}a^{7}+\frac{11}{57}a^{6}-\frac{31}{19}a^{5}-\frac{142}{57}a^{4}+\frac{160}{19}a^{3}+\frac{230}{19}a^{2}-\frac{284}{19}a-\frac{527}{19}$, $\frac{6}{19}a^{7}+\frac{1}{19}a^{6}-\frac{131}{19}a^{5}+\frac{32}{19}a^{4}+\frac{936}{19}a^{3}-\frac{680}{19}a^{2}-\frac{2182}{19}a+\frac{2879}{19}$, $\frac{111}{19}a^{7}-\frac{293}{19}a^{6}-\frac{1445}{19}a^{5}+\frac{3810}{19}a^{4}+\frac{3902}{19}a^{3}-\frac{10254}{19}a^{2}-\frac{7440}{19}a+\frac{19585}{19}$
|
| |
| Regulator: | \( 16649.8325698 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 16649.8325698 \cdot 2}{2\cdot\sqrt{126142572920832}}\cr\approx \mathstrut & 0.936394269966 \end{aligned}\]
Galois group
$C_2\wr C_4$ (as 8T28):
| A solvable group of order 64 |
| The 13 conjugacy class representatives for $(((C_4 \times C_2): C_2):C_2):C_2$ |
| Character table for $(((C_4 \times C_2): C_2):C_2):C_2$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 4.2.32448.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 8.2.4919560343912448.10 |
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 | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }$ | R | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.22a1.72 | $x^{8} + 12 x^{7} + 42 x^{6} + 88 x^{5} + 127 x^{4} + 128 x^{3} + 102 x^{2} + 76 x + 23$ | $4$ | $2$ | $22$ | $C_8$ | $$[3, 4]^{2}$$ |
|
\(3\)
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 3.1.4.3a1.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ | |
|
\(13\)
| 13.1.2.1a1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 13.1.2.1a1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 13.1.4.3a1.3 | $x^{4} + 52$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |