Normalized defining polynomial
\( x^{5} - 545x^{2} - 1635x - 1308 \)
Invariants
| Degree: | $5$ |
| |
| Signature: | $(1, 2)$ |
| |
| Discriminant: |
\(1429226380125\)
\(\medspace = 3^{4}\cdot 5^{3}\cdot 109^{4}\)
|
| |
| Root discriminant: | \(269.79\) |
| |
| Galois root discriminant: | $3^{7/6}5^{5/6}109^{4/5}\approx 587.5673647942456$ | ||
| Ramified primes: |
\(3\), \(5\), \(109\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{5}a^{2}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{3}-\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{25}a^{4}-\frac{2}{25}a^{3}-\frac{1}{25}a^{2}+\frac{2}{25}a+\frac{1}{25}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{14\cdots 04}{25}a^{4}-\frac{12\cdots 68}{25}a^{3}+\frac{33\cdots 66}{25}a^{2}-\frac{76\cdots 92}{25}a-\frac{16\cdots 31}{25}$, $\frac{13\cdots 66}{25}a^{4}-\frac{42\cdots 22}{25}a^{3}+\frac{95\cdots 74}{25}a^{2}-\frac{75\cdots 78}{25}a+\frac{15\cdots 41}{25}$
|
| |
| Regulator: | \( 55162.1290118 \) |
| |
| Unit signature rank: | \( 1 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{2}\cdot 55162.1290118 \cdot 1}{2\cdot\sqrt{1429226380125}}\cr\approx \mathstrut & 1.8215883687 \end{aligned}\]
Galois group
| A non-solvable group of order 120 |
| The 7 conjugacy class representatives for $S_5$ |
| Character table for $S_5$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 6 sibling: | deg 6 |
| Degree 10 siblings: | deg 10, deg 10 |
| Degree 12 sibling: | deg 12 |
| Degree 15 sibling: | deg 15 |
| Degree 20 siblings: | deg 20, deg 20, deg 20 |
| Degree 24 sibling: | deg 24 |
| Degree 30 siblings: | deg 30, deg 30, deg 30 |
| Degree 40 sibling: | deg 40 |
| 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 | ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | R | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.3.3a1.1 | $x^{3} + 3 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $$[\frac{3}{2}]_{2}$$ | |
|
\(5\)
| 5.1.2.1a1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.1.3.2a1.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
|
\(109\)
| 109.1.5.4a1.1 | $x^{5} + 109$ | $5$ | $1$ | $4$ | $D_{5}$ | $$[\ ]_{5}^{2}$$ |