Normalized defining polynomial
\( x^{8} + 4x^{6} - 18x^{4} - 15x^{2} + 5 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(22632992000\)
\(\medspace = 2^{8}\cdot 5^{3}\cdot 29^{4}\)
|
| |
| Root discriminant: | \(19.69\) |
| |
| Galois root discriminant: | $2\cdot 5^{3/4}29^{1/2}\approx 36.01276755471423$ | ||
| Ramified primes: |
\(2\), \(5\), \(29\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_4$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{77}a^{6}+\frac{20}{77}a^{4}-\frac{6}{77}a^{2}-\frac{34}{77}$, $\frac{1}{77}a^{7}+\frac{20}{77}a^{5}-\frac{6}{77}a^{3}-\frac{34}{77}a$
| 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: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{9}{77}a^{6}+\frac{26}{77}a^{4}-\frac{208}{77}a^{2}+\frac{2}{77}$, $\frac{10}{77}a^{6}+\frac{46}{77}a^{4}-\frac{137}{77}a^{2}-\frac{186}{77}$, $\frac{5}{77}a^{6}+\frac{23}{77}a^{4}-\frac{30}{77}a^{2}-\frac{16}{77}$, $\frac{7}{11}a^{7}-\frac{25}{77}a^{6}+\frac{30}{11}a^{5}-\frac{115}{77}a^{4}-\frac{119}{11}a^{3}+\frac{381}{77}a^{2}-\frac{150}{11}a+\frac{542}{77}$, $\frac{20}{77}a^{7}+\frac{1}{7}a^{6}+\frac{92}{77}a^{5}+\frac{6}{7}a^{4}-\frac{274}{77}a^{3}-\frac{6}{7}a^{2}-\frac{295}{77}a-\frac{13}{7}$
|
| |
| Regulator: | \( 241.861188917 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 241.861188917 \cdot 1}{2\cdot\sqrt{22632992000}}\cr\approx \mathstrut & 0.507744142744 \end{aligned}\]
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
| \(\Q(\sqrt{29}) \), 4.4.4205.1 |
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: | 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 | ${\href{/padicField/3.8.0.1}{8} }$ | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.2.8a1.2 | $x^{8} + 2 x^{5} + 4 x^{4} + 4 x^{3} + x^{2} + 4 x + 5$ | $2$ | $4$ | $8$ | $C_8$ | $$[2]^{4}$$ |
|
\(5\)
| 5.1.4.3a1.3 | $x^{4} + 15$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.4.1.0a1.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
|
\(29\)
| 29.4.2.4a1.2 | $x^{8} + 4 x^{6} + 30 x^{5} + 8 x^{4} + 60 x^{3} + 233 x^{2} + 60 x + 33$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *32 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.145.2t1.a.a | $1$ | $ 5 \cdot 29 $ | \(\Q(\sqrt{145}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *32 | 1.29.2t1.a.a | $1$ | $ 29 $ | \(\Q(\sqrt{29}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.5.4t1.a.a | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ | |
| 1.145.4t1.b.a | $1$ | $ 5 \cdot 29 $ | 4.0.105125.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
| 1.5.4t1.a.b | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ | |
| 1.145.4t1.b.b | $1$ | $ 5 \cdot 29 $ | 4.0.105125.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
| 2.725.4t3.a.a | $2$ | $ 5^{2} \cdot 29 $ | 4.0.3625.1 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
| *32 | 2.145.4t3.b.a | $2$ | $ 5 \cdot 29 $ | 4.4.725.1 | $D_{4}$ (as 4T3) | $1$ | $2$ |
| 2.11600.8t17.e.a | $2$ | $ 2^{4} \cdot 5^{2} \cdot 29 $ | 8.4.22632992000.1 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ | |
| 2.11600.8t17.e.b | $2$ | $ 2^{4} \cdot 5^{2} \cdot 29 $ | 8.4.22632992000.1 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ | |
| *32 | 2.2320.8t17.g.a | $2$ | $ 2^{4} \cdot 5 \cdot 29 $ | 8.4.22632992000.1 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ |
| *32 | 2.2320.8t17.g.b | $2$ | $ 2^{4} \cdot 5 \cdot 29 $ | 8.4.22632992000.1 | $C_4\wr C_2$ (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 *.