Normalized defining polynomial
\( x^{3} - 1742231510415 \)
Invariants
| Degree: | $3$ |
| |
| Signature: | $[1, 1]$ |
| |
| Discriminant: |
\(-32787760398261603075\)
\(\medspace = -\,3^{5}\cdot 5^{2}\cdot 11^{2}\cdot 17^{2}\cdot 19^{2}\cdot 23^{2}\cdot 29^{2}\cdot 31^{2}\)
|
| |
| Root discriminant: | \($3\,200\,643$.11\) |
| |
| Galois root discriminant: | $3^{11/6}5^{2/3}11^{2/3}17^{2/3}19^{2/3}23^{2/3}29^{2/3}31^{2/3}\approx 3843770.594580035$ | ||
| Ramified primes: |
\(3\), \(5\), \(11\), \(17\), \(19\), \(23\), \(29\), \(31\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{1581}a^{2}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | not computed |
| |
| Narrow class group: | not computed |
|
Unit group
| Rank: | $1$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental unit: | not computed |
| |
| Regulator: | not computed |
|
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 = \mathstrut &\frac{2^{1}\cdot(2\pi)^{1}\cdot R \cdot h}{2\cdot\sqrt{32787760398261603075}}\cr\mathstrut & \text{
Galois group
| A solvable group of order 6 |
| The 3 conjugacy class representatives for $S_3$ |
| Character table for $S_3$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Galois closure: | deg 6 |
| 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 | ${\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | R | ${\href{/padicField/7.3.0.1}{3} }$ | R | ${\href{/padicField/13.3.0.1}{3} }$ | R | R | R | R | R | ${\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.3.5a1.2 | $x^{3} + 9 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $$[\frac{5}{2}]_{2}$$ |
|
\(5\)
| 5.1.3.2a1.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
|
\(11\)
| 11.1.3.2a1.1 | $x^{3} + 11$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
|
\(17\)
| 17.1.3.2a1.1 | $x^{3} + 17$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
|
\(19\)
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
|
\(23\)
| 23.1.3.2a1.1 | $x^{3} + 23$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
|
\(29\)
| 29.1.3.2a1.1 | $x^{3} + 29$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
|
\(31\)
| 31.1.3.2a1.1 | $x^{3} + 31$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |