Normalized defining polynomial
\( x^{8} + 4x^{6} - 40x^{5} - 48x^{4} + 200x^{3} + 152x^{2} - 216x - 142 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(2066969788416\)
\(\medspace = 2^{24}\cdot 3^{6}\cdot 13^{2}\)
|
| |
| Root discriminant: | \(34.63\) |
| |
| Galois root discriminant: | $2^{51/16}3^{25/18}13^{1/2}\approx 151.06824943181383$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{269403}a^{7}+\frac{27400}{269403}a^{6}+\frac{37815}{89801}a^{5}-\frac{22885}{89801}a^{4}+\frac{31367}{89801}a^{3}-\frac{88513}{269403}a^{2}-\frac{90242}{269403}a+\frac{13173}{89801}$
| 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: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{190}{89801}a^{7}-\frac{2458}{89801}a^{6}+\frac{2310}{89801}a^{5}-\frac{23305}{89801}a^{4}+\frac{98592}{89801}a^{3}+\frac{65118}{89801}a^{2}-\frac{263392}{89801}a-\frac{124475}{89801}$, $\frac{26149}{269403}a^{7}-\frac{13193}{89801}a^{6}+\frac{166673}{269403}a^{5}-\frac{435006}{89801}a^{4}+\frac{242752}{89801}a^{3}+\frac{3955781}{269403}a^{2}-\frac{2012803}{269403}a-\frac{2382403}{269403}$, $\frac{16129}{269403}a^{7}+\frac{23879}{269403}a^{6}+\frac{148831}{269403}a^{5}-\frac{119856}{89801}a^{4}-\frac{289894}{89801}a^{3}+\frac{748529}{269403}a^{2}+\frac{323067}{89801}a+\frac{174055}{269403}$, $\frac{30607}{269403}a^{7}-\frac{19739}{269403}a^{6}+\frac{48417}{89801}a^{5}-\frac{442400}{89801}a^{4}-\frac{192324}{89801}a^{3}+\frac{5926043}{269403}a^{2}+\frac{421468}{269403}a-\frac{1996101}{89801}$, $\frac{447245}{269403}a^{7}-\frac{809072}{269403}a^{6}+\frac{3017657}{269403}a^{5}-\frac{7765935}{89801}a^{4}+\frac{6397566}{89801}a^{3}+\frac{64545064}{269403}a^{2}-\frac{14378843}{89801}a-\frac{44678563}{269403}$
|
| |
| Regulator: | \( 3724.5762581 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 3724.5762581 \cdot 1}{2\cdot\sqrt{2066969788416}}\cr\approx \mathstrut & 0.81820010304 \end{aligned}\]
Galois group
$\POPlus(4,3)$ (as 8T45):
| A solvable group of order 576 |
| The 16 conjugacy class representatives for $(A_4\wr C_2):C_2$ |
| Character table for $(A_4\wr C_2):C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 36 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 | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.24b2.1 | $x^{8} + 4 x^{6} + 8 x + 2$ | $8$ | $1$ | $24$ | $C_2^4:C_6$ | $$[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{3}$$ |
|
\(3\)
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 3.2.3.6a4.1 | $x^{6} + 6 x^{5} + 18 x^{4} + 38 x^{3} + 54 x^{2} + 48 x + 23$ | $3$ | $2$ | $6$ | $S_3^2$ | $$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$$ | |
|
\(13\)
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 13.2.2.2a1.2 | $x^{4} + 24 x^{3} + 148 x^{2} + 48 x + 17$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |