Normalized defining polynomial
\( x^{8} - x^{7} + x^{6} + 2x^{5} - 3x^{4} - 2x^{3} + x^{2} + x + 1 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(16471125\)
\(\medspace = 3^{2}\cdot 5^{3}\cdot 11^{4}\)
|
| |
| Root discriminant: | \(7.98\) |
| |
| Galois root discriminant: | $3^{1/2}5^{3/4}11^{1/2}\approx 19.208102881010017$ | ||
| Ramified primes: |
\(3\), \(5\), \(11\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_4$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-11}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$
| 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: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{2}{3}a^{5}+\frac{1}{3}a^{3}-a^{2}+\frac{2}{3}$, $a$, $\frac{1}{3}a^{7}-\frac{2}{3}a^{6}+\frac{1}{3}a^{5}+\frac{2}{3}a^{4}-\frac{7}{3}a^{3}+\frac{1}{3}a^{2}+\frac{5}{3}a$
|
| |
| Regulator: | \( 1.69679589997 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 1.69679589997 \cdot 1}{2\cdot\sqrt{16471125}}\cr\approx \mathstrut & 0.325804780469 \end{aligned}\]
Galois group
$C_4\wr C_2$ (as 8T17):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-11}) \), 4.0.605.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 8 sibling: | data not computed |
| Degree 16 siblings: | 16.4.2837698174072265625.2, 16.0.169561224228515625.2 |
| 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 | ${\href{/padicField/2.8.0.1}{8} }$ | R | R | ${\href{/padicField/7.8.0.1}{8} }$ | R | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 3.2.2.2a1.1 | $x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ | |
|
\(5\)
| 5.1.4.3a1.3 | $x^{4} + 15$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.4.1.0a1.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
|
\(11\)
| 11.4.2.4a1.2 | $x^{8} + 16 x^{6} + 20 x^{5} + 68 x^{4} + 160 x^{3} + 132 x^{2} + 40 x + 15$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *32 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.55.2t1.a.a | $1$ | $ 5 \cdot 11 $ | \(\Q(\sqrt{-55}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *32 | 1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.15.4t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ | |
| 1.15.4t1.a.b | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ | |
| 1.165.4t1.a.a | $1$ | $ 3 \cdot 5 \cdot 11 $ | 4.0.136125.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
| 1.165.4t1.a.b | $1$ | $ 3 \cdot 5 \cdot 11 $ | 4.0.136125.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
| 2.2475.4t3.c.a | $2$ | $ 3^{2} \cdot 5^{2} \cdot 11 $ | 4.2.12375.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| *32 | 2.55.4t3.c.a | $2$ | $ 5 \cdot 11 $ | 4.2.275.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
| 2.825.8t17.d.a | $2$ | $ 3 \cdot 5^{2} \cdot 11 $ | 8.0.16471125.2 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ | |
| *32 | 2.165.8t17.d.a | $2$ | $ 3 \cdot 5 \cdot 11 $ | 8.0.16471125.2 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ |
| 2.825.8t17.d.b | $2$ | $ 3 \cdot 5^{2} \cdot 11 $ | 8.0.16471125.2 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ | |
| *32 | 2.165.8t17.d.b | $2$ | $ 3 \cdot 5 \cdot 11 $ | 8.0.16471125.2 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ |
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 *.