Normalized defining polynomial
\( x^{3} - 77120715x - 474789795970 \)
Invariants
| Degree: | $3$ |
| |
| Signature: | $[1, 1]$ |
| |
| Discriminant: |
\(-996109592817061800\)
\(\medspace = -\,2^{3}\cdot 3^{4}\cdot 5^{2}\cdot 7^{2}\cdot 11\cdot 17\cdot 19^{2}\cdot 23\cdot 29^{2}\cdot 31^{2}\)
|
| |
| Root discriminant: | \($998\,701$.51\) |
| |
| Galois root discriminant: | $2^{3/2}3^{4/3}5^{2/3}7^{2/3}11^{1/2}17^{1/2}19^{2/3}23^{1/2}29^{2/3}31^{2/3}\approx 5695680.407167414$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\), \(11\), \(17\), \(19\), \(23\), \(29\), \(31\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-8602}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{2066}a^{2}+\frac{759}{2066}a+\frac{333}{1033}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | not computed |
| |
| Narrow class group: | not computed |
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Unit group
| Rank: | $1$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental unit: | not computed |
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| Regulator: | not computed |
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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{996109592817061800}}\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 | R | R | R | R | R | ${\href{/padicField/13.3.0.1}{3} }$ | R | R | R | R | R | ${\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 2.1.2.3a1.2 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ | |
|
\(3\)
| 3.1.3.4a1.1 | $x^{3} + 3 x^{2} + 3$ | $3$ | $1$ | $4$ | $S_3$ | $$[2]^{2}$$ |
|
\(5\)
| 5.1.3.2a1.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
|
\(7\)
| 7.1.3.2a1.1 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
|
\(11\)
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 11.1.2.1a1.1 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(17\)
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 17.1.2.1a1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(19\)
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
|
\(23\)
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 23.1.2.1a1.1 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(29\)
| 29.1.3.2a1.1 | $x^{3} + 29$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
|
\(31\)
| 31.1.3.2a1.3 | $x^{3} + 279$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |