Normalized defining polynomial
\( x^{8} + 4x^{6} - 26x^{4} - 40x^{2} + 20 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(11023155200000\)
\(\medspace = 2^{22}\cdot 5^{5}\cdot 29^{2}\)
|
| |
| Root discriminant: | \(42.69\) |
| |
| Galois root discriminant: | $2^{11/4}5^{3/4}29^{1/2}\approx 121.1320285604979$ | ||
| Ramified primes: |
\(2\), \(5\), \(29\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{46}a^{6}-\frac{4}{23}a^{4}-\frac{11}{23}a^{2}-\frac{3}{23}$, $\frac{1}{46}a^{7}-\frac{4}{23}a^{5}-\frac{11}{23}a^{3}-\frac{3}{23}a$
| 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: | $C_{4}$, which has order $4$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{23}a^{6}+\frac{7}{46}a^{4}-\frac{22}{23}a^{2}-\frac{29}{23}$, $\frac{1}{23}a^{7}+\frac{7}{46}a^{5}-\frac{22}{23}a^{3}-\frac{29}{23}a-1$, $\frac{1}{23}a^{7}+\frac{7}{46}a^{5}-\frac{22}{23}a^{3}-\frac{29}{23}a+1$, $\frac{1}{46}a^{6}-\frac{4}{23}a^{4}-\frac{11}{23}a^{2}-\frac{3}{23}$, $\frac{1447}{23}a^{7}-\frac{2356}{23}a^{6}+\frac{10228}{23}a^{5}-\frac{18274}{23}a^{4}-\frac{1888}{23}a^{3}-\frac{10866}{23}a^{2}-\frac{26254}{23}a+\frac{23497}{23}$
|
| |
| Regulator: | \( 1625.22440994 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 1625.22440994 \cdot 2}{2\cdot\sqrt{11023155200000}}\cr\approx \mathstrut & 0.309200434040 \end{aligned}\]
Galois group
$C_2\wr C_4$ (as 8T28):
| A solvable group of order 64 |
| The 13 conjugacy class representatives for $(((C_4 \times C_2): C_2):C_2):C_2$ |
| Character table for $(((C_4 \times C_2): C_2):C_2):C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.46400.1 |
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: | 8.0.1900544000000.3 |
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 | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.22a1.72 | $x^{8} + 12 x^{7} + 42 x^{6} + 88 x^{5} + 127 x^{4} + 128 x^{3} + 102 x^{2} + 76 x + 23$ | $4$ | $2$ | $22$ | $C_8$ | $$[3, 4]^{2}$$ |
|
\(5\)
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(29\)
| 29.2.1.0a1.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 29.2.1.0a1.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 29.2.2.2a1.2 | $x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |