Normalized defining polynomial
\( x^{8} + 24x^{6} + 272x^{4} + 1472x^{2} + 4232 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(0, 4)$ |
| |
| Discriminant: |
\(10224169648128\)
\(\medspace = 2^{31}\cdot 3^{2}\cdot 23^{2}\)
|
| |
| Root discriminant: | \(42.29\) |
| |
| Galois root discriminant: | $2^{141/32}3^{1/2}23^{1/2}\approx 176.13187718534644$ | ||
| Ramified primes: |
\(2\), \(3\), \(23\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-2}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2300}a^{6}+\frac{81}{1150}a^{4}-\frac{93}{575}a^{2}+\frac{8}{25}$, $\frac{1}{2300}a^{7}+\frac{81}{1150}a^{5}-\frac{93}{575}a^{3}+\frac{8}{25}a$
| 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: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{7}{1150}a^{6}-\frac{8}{575}a^{4}-\frac{152}{575}a^{2}-\frac{63}{25}$, $\frac{12}{575}a^{7}-\frac{2}{25}a^{6}+\frac{219}{575}a^{5}-\frac{24}{25}a^{4}+\frac{1286}{575}a^{3}-\frac{156}{25}a^{2}+\frac{184}{25}a-\frac{297}{25}$, $\frac{50653}{460}a^{7}-\frac{19187}{575}a^{6}+\frac{464523}{230}a^{5}+\frac{904787}{1150}a^{4}+\frac{1774926}{115}a^{3}+\frac{10352389}{575}a^{2}+\frac{123484}{5}a+\frac{3507941}{25}$
|
| |
| Regulator: | \( 1805.2994718 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 1805.2994718 \cdot 2}{2\cdot\sqrt{10224169648128}}\cr\approx \mathstrut & 0.87994333267 \end{aligned}\]
Galois group
$C_2\wr D_4$ (as 8T35):
| A solvable group of order 128 |
| The 20 conjugacy class representatives for $C_2 \wr C_2\wr C_2$ |
| Character table for $C_2 \wr C_2\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-2 + \sqrt{-2}})\) |
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: | 8.0.10224169648128.5 |
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 | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{6}$ | R | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.31a1.62 | $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 8 x^{2} + 50$ | $8$ | $1$ | $31$ | $C_2 \wr C_2\wr C_2$ | $$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$$ |
|
\(3\)
| 3.4.1.0a1.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(23\)
| 23.2.2.2a1.1 | $x^{4} + 42 x^{3} + 451 x^{2} + 233 x + 25$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
| 23.4.1.0a1.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |