Normalized defining polynomial
\( x^{8} + 16x^{6} - 164x^{4} - 3072x^{2} - 10092 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-4919560343912448\)
\(\medspace = -\,2^{22}\cdot 3^{5}\cdot 13^{6}\)
|
| |
| Root discriminant: | \(91.51\) |
| |
| Galois root discriminant: | $2^{11/4}3^{3/4}13^{3/4}\approx 104.98589220070072$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{8}a^{4}-\frac{1}{2}a^{2}+\frac{1}{4}$, $\frac{1}{8}a^{5}-\frac{1}{2}a^{3}+\frac{1}{4}a$, $\frac{1}{408}a^{6}-\frac{25}{408}a^{4}+\frac{33}{68}a^{2}+\frac{5}{68}$, $\frac{1}{11832}a^{7}-\frac{535}{11832}a^{5}-\frac{443}{1972}a^{3}+\frac{39}{1972}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{4}$, which has order $8$ |
| |
| Narrow class group: | $C_{2}\times C_{2}\times C_{4}$, which has order $16$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{136}a^{6}+\frac{9}{136}a^{4}-\frac{105}{68}a^{2}-\frac{631}{68}$, $\frac{2}{51}a^{6}+\frac{1}{51}a^{4}-\frac{106}{17}a^{2}-\frac{473}{17}$, $\frac{1}{408}a^{6}-\frac{25}{408}a^{4}-\frac{103}{68}a^{2}-\frac{471}{68}$, $\frac{16\cdots 85}{5916}a^{7}-\frac{25\cdots 87}{24}a^{6}+\frac{12\cdots 73}{1479}a^{5}-\frac{78\cdots 15}{24}a^{4}+\frac{74\cdots 33}{986}a^{3}-\frac{11\cdots 79}{4}a^{2}+\frac{10\cdots 96}{493}a-\frac{30\cdots 01}{4}$
|
| |
| Regulator: | \( 20336.8803799 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 20336.8803799 \cdot 8}{2\cdot\sqrt{4919560343912448}}\cr\approx \mathstrut & 1.15074944963 \end{aligned}\]
Galois group
$C_2\wr C_4$ (as 8T27):
| A solvable group of order 64 |
| The 13 conjugacy class representatives for $((C_8 : C_2):C_2):C_2$ |
| Character table for $((C_8 : C_2):C_2):C_2$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 4.4.1265472.2 |
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: | 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 | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }$ | R | ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{6}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 | $x^{8} + 12 x^{7} + 42 x^{6} + 88 x^{5} + 127 x^{4} + 128 x^{3} + 102 x^{2} + 76 x + 23$ | $4$ | $2$ | $22$ | $C_8$ | $$[3, 4]^{2}$$ |
|
\(3\)
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.1.4.3a1.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ | |
|
\(13\)
| 13.1.4.3a1.3 | $x^{4} + 52$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 13.1.4.3a1.3 | $x^{4} + 52$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |