Normalized defining polynomial
\( x^{8} + 24x^{6} - 3336x^{4} - 64960x^{2} + 269120 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(9270473523200000\)
\(\medspace = 2^{22}\cdot 5^{5}\cdot 29^{4}\)
|
| |
| Root discriminant: | \(99.06\) |
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{4}a^{3}$, $\frac{1}{8}a^{4}$, $\frac{1}{8}a^{5}$, $\frac{1}{8611376}a^{6}-\frac{27373}{1076422}a^{4}-\frac{113381}{538211}a^{2}-\frac{3411}{18559}$, $\frac{1}{17222752}a^{7}-\frac{27373}{2152844}a^{5}-\frac{113381}{1076422}a^{3}+\frac{7574}{18559}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}\times C_{2}$, which has order $8$ (assuming GRH) |
| |
| Narrow class group: | $C_{2}\times C_{2}\times C_{4}$, which has order $16$ (assuming GRH) |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{5}{148472}a^{6}+\frac{61}{148472}a^{4}-\frac{1711}{18559}a^{2}-\frac{24122}{18559}$, $\frac{177}{4305688}a^{6}-\frac{1122}{538211}a^{4}-\frac{80309}{1076422}a^{2}-\frac{1159}{18559}$, $\frac{575}{8611376}a^{6}+\frac{12787}{4305688}a^{4}-\frac{70544}{538211}a^{2}-\frac{68307}{18559}$, $\frac{4913829}{8611376}a^{7}+\frac{159057}{148472}a^{6}+\frac{8453589}{538211}a^{5}+\frac{1094787}{37118}a^{4}-\frac{1989383823}{1076422}a^{3}-\frac{64384353}{18559}a^{2}-\frac{808775064}{18559}a-\frac{1518210901}{18559}$, $\frac{60519492425}{8611376}a^{7}+\frac{225520696967}{4305688}a^{6}+\frac{2407001026967}{4305688}a^{5}+\frac{4484741503167}{1076422}a^{4}+\frac{16374763251951}{2152844}a^{3}+\frac{30509578537590}{538211}a^{2}-\frac{631947691897}{18559}a-\frac{4709795930821}{18559}$
|
| |
| Regulator: | \( 17213.0520607 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 17213.0520607 \cdot 8}{2\cdot\sqrt{9270473523200000}}\cr\approx \mathstrut & 0.451696404956 \end{aligned}\] (assuming GRH)
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 | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(29\)
| 29.1.2.1a1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 29.1.2.1a1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 29.2.2.2a1.2 | $x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |