Normalized defining polynomial
\( x^{8} - 900x^{4} + 12000x^{2} - 67500 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-464380231680000\)
\(\medspace = -\,2^{22}\cdot 3^{11}\cdot 5^{4}\)
|
| |
| Root discriminant: | \(68.13\) |
| |
| Galois root discriminant: | $2^{11/4}3^{11/6}5^{1/2}\approx 112.73007360429621$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{5}a^{2}$, $\frac{1}{5}a^{3}$, $\frac{1}{200}a^{4}-\frac{1}{10}a^{2}+\frac{1}{4}$, $\frac{1}{200}a^{5}-\frac{1}{10}a^{3}+\frac{1}{4}a$, $\frac{1}{3000}a^{6}-\frac{1}{20}a^{2}$, $\frac{1}{9000}a^{7}+\frac{1}{20}a^{3}+\frac{1}{3}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}$, which has order $4$ |
| |
| Narrow class group: | $C_{2}\times C_{4}$, which has order $8$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{1500}a^{6}+\frac{1}{200}a^{4}-\frac{3}{5}a^{2}+\frac{13}{4}$, $\frac{1}{200}a^{4}+\frac{1}{10}a^{2}-\frac{11}{4}$, $\frac{88}{125}a^{7}+\frac{9971}{3000}a^{6}+\frac{3139}{200}a^{5}+\frac{593}{8}a^{4}-\frac{567}{2}a^{3}-\frac{26753}{20}a^{2}+\frac{8535}{4}a+\frac{40273}{4}$, $\frac{111273530129101}{250}a^{7}-\frac{89231302926289}{300}a^{6}-\frac{240679479304083}{100}a^{5}+16331766779850a^{4}-\frac{23\cdots 98}{5}a^{3}+\frac{10\cdots 41}{2}a^{2}+\frac{13\cdots 77}{2}a-16\!\cdots\!91$
|
| |
| Regulator: | \( 15567.3550861 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 15567.3550861 \cdot 4}{2\cdot\sqrt{464380231680000}}\cr\approx \mathstrut & 1.43353256063 \end{aligned}\]
Galois group
$\GL(2,3)$ (as 8T23):
| A solvable group of order 48 |
| The 8 conjugacy class representatives for $\textrm{GL(2,3)}$ |
| Character table for $\textrm{GL(2,3)}$ |
Intermediate fields
| 4.2.15552.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 16 sibling: | deg 16 |
| Degree 24 sibling: | deg 24 |
| Arithmetically equivalent sibling: | 8.2.464380231680000.30 |
| Minimal sibling: | 8.2.464380231680000.30 |
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 | R | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.22a1.72 | $x^{8} + 12 x^{7} + 42 x^{6} + 88 x^{5} + 127 x^{4} + 128 x^{3} + 102 x^{2} + 76 x + 23$ | $4$ | $2$ | $22$ | $C_8$ | $$[3, 4]^{2}$$ |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 3.1.6.11a1.1 | $x^{6} + 3$ | $6$ | $1$ | $11$ | $S_3$ | $$[\frac{5}{2}]_{2}$$ | |
|
\(5\)
| 5.4.2.4a1.1 | $x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 21 x + 4$ | $2$ | $4$ | $4$ | $C_8$ | $$[\ ]_{2}^{4}$$ |