Normalized defining polynomial
\( x^{8} - 24x^{4} - 3 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(2, 3)$ |
| |
| Discriminant: |
\(-36691771392\)
\(\medspace = -\,2^{24}\cdot 3^{7}\)
|
| |
| Root discriminant: | \(20.92\) |
| |
| Galois root discriminant: | $2^{25/8}3^{7/8}\approx 22.813915798899377$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{7}a^{4}+\frac{2}{7}$, $\frac{1}{7}a^{5}+\frac{2}{7}a$, $\frac{1}{7}a^{6}+\frac{2}{7}a^{2}$, $\frac{1}{7}a^{7}+\frac{2}{7}a^{3}$
| 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: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{7}a^{4}+\frac{2}{7}$, $\frac{1}{7}a^{7}-\frac{26}{7}a^{3}+a-1$, $\frac{1}{7}a^{7}-\frac{26}{7}a^{3}+a+1$, $\frac{3}{7}a^{6}-\frac{3}{7}a^{5}+\frac{3}{7}a^{4}-a^{3}-\frac{57}{7}a^{2}+\frac{50}{7}a-\frac{22}{7}$
|
| |
| Regulator: | \( 322.773113924 \) |
| |
| Unit signature rank: | \( 1 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 322.773113924 \cdot 1}{2\cdot\sqrt{36691771392}}\cr\approx \mathstrut & 0.83595444873 \end{aligned}\]
Galois group
| A solvable group of order 32 |
| The 11 conjugacy class representatives for $Z_8 : Z_8^\times$ |
| Character table for $Z_8 : Z_8^\times$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), \(\Q(\sqrt[4]{3})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 32 |
| Degree 16 siblings: | 16.0.1346286087882789617664.44, 16.4.5385144351531158470656.12, 16.0.1346286087882789617664.30, 16.0.5385144351531158470656.27 |
| Arithmetically equivalent sibling: | 8.2.36691771392.2 |
| 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 | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.24c1.33 | $x^{8} + 2 x^{4} + 4 x^{2} + 8 x + 30$ | $8$ | $1$ | $24$ | $Z_8 : Z_8^\times$ | $$[2, 2, 3, 4]^{2}$$ |
|
\(3\)
| 3.1.8.7a1.2 | $x^{8} + 6$ | $8$ | $1$ | $7$ | $QD_{16}$ | $$[\ ]_{8}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *32 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *32 | 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.24.2t1.b.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{-6}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.24.2t1.a.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{6}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *32 | 2.576.4t3.a.a | $2$ | $ 2^{6} \cdot 3^{2}$ | \(\Q(\sqrt{3 + \sqrt{-3}})\) | $D_{4}$ (as 4T3) | $1$ | $0$ |
| 2.144.4t3.b.a | $2$ | $ 2^{4} \cdot 3^{2}$ | \(\Q(\sqrt{6 +2 \sqrt{-3}})\) | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| *32 | 4.5308416.8t15.a.a | $4$ | $ 2^{16} \cdot 3^{4}$ | 8.2.36691771392.1 | $Z_8 : Z_8^\times$ (as 8T15) | $1$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.