Normalized defining polynomial
\( x^{8} - 12x^{6} + 66x^{4} + 72x^{2} + 18 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(0, 4)$ |
| |
| Discriminant: |
\(97844723712\)
\(\medspace = 2^{27}\cdot 3^{6}\)
|
| |
| Root discriminant: | \(23.65\) |
| |
| Galois root discriminant: | $2^{29/8}3^{3/4}\approx 28.12384367863563$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| 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: | \(\Q(\sqrt{-1}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{15}a^{6}+\frac{2}{15}a^{4}-\frac{2}{5}a^{2}+\frac{1}{5}$, $\frac{1}{15}a^{7}+\frac{2}{15}a^{5}-\frac{2}{5}a^{3}+\frac{1}{5}a$
| 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: |
\( \frac{2}{3} a^{6} - \frac{25}{3} a^{4} + 48 a^{2} + 25 \)
(order $4$)
|
| |
| Fundamental units: |
$\frac{6}{5}a^{6}-\frac{224}{15}a^{4}+\frac{429}{5}a^{2}+\frac{243}{5}$, $\frac{2}{3}a^{7}-\frac{2}{15}a^{6}-\frac{25}{3}a^{5}+\frac{26}{15}a^{4}+48a^{3}-\frac{51}{5}a^{2}+26a-\frac{27}{5}$, $654a^{7}-\frac{691}{3}a^{6}-\frac{24503}{3}a^{5}+\frac{8599}{3}a^{4}+47156a^{3}-16475a^{2}+24042a-9269$
|
| |
| Regulator: | \( 366.239918759 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 366.239918759 \cdot 2}{4\cdot\sqrt{97844723712}}\cr\approx \mathstrut & 0.91240246756 \end{aligned}\]
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
| \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3 +3 i})\) |
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: | 16.0.153177439332441840943104.166, 16.0.9573589958277615058944.70, 16.0.153177439332441840943104.169, 16.4.153177439332441840943104.88 |
| Arithmetically equivalent sibling: | 8.0.97844723712.6 |
| 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 | ${\href{/padicField/5.2.0.1}{2} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.27a1.29 | $x^{8} + 4 x^{6} + 18 x^{4} + 8 x^{2} + 16 x + 2$ | $8$ | $1$ | $27$ | $Z_8 : Z_8^\times$ | $$[2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$$ |
|
\(3\)
| 3.2.4.6a1.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 16$ | $4$ | $2$ | $6$ | $C_8:C_2$ | $$[\ ]_{4}^{4}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *32 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 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) | $1$ | $-1$ | |
| 1.24.2t1.b.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{-6}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.24.2t1.a.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{6}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *32 | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *32 | 2.1152.4t3.f.a | $2$ | $ 2^{7} \cdot 3^{2}$ | \(\Q(\sqrt{3 +3 \sqrt{2}})\) | $D_{4}$ (as 4T3) | $1$ | $0$ |
| 2.128.4t3.c.a | $2$ | $ 2^{7}$ | \(\Q(\sqrt{1 + \sqrt{2}})\) | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| *32 | 4.21233664.8t15.k.a | $4$ | $ 2^{18} \cdot 3^{4}$ | 8.0.97844723712.2 | $Z_8 : Z_8^\times$ (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 *.