Normalized defining polynomial
\( x^{5} - 2x^{4} + 67x^{3} + 2284x^{2} + 15479x + 24538 \)
Invariants
| Degree: | $5$ |
| |
| Signature: | $(1, 2)$ |
| |
| Discriminant: |
\(491371558441\)
\(\medspace = 59^{2}\cdot 109^{4}\)
|
| |
| Root discriminant: | \(217.91\) |
| |
| Galois root discriminant: | $59^{1/2}109^{4/5}\approx 327.61758465000895$ | ||
| Ramified primes: |
\(59\), \(109\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{90206}a^{4}-\frac{119}{90206}a^{3}-\frac{31113}{90206}a^{2}-\frac{5419}{45103}a-\frac{12231}{45103}$
| Monogenic: | No | |
| Index: | $2$ | |
| Inessential primes: | $2$ |
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: |
$\frac{13\cdots 18}{45103}a^{4}-\frac{17\cdots 21}{90206}a^{3}+\frac{22\cdots 93}{90206}a^{2}+\frac{58\cdots 39}{90206}a+\frac{59\cdots 80}{45103}$, $\frac{50\cdots 32}{45103}a^{4}-\frac{12\cdots 46}{45103}a^{3}+\frac{16\cdots 42}{45103}a^{2}-\frac{68\cdots 73}{45103}a-\frac{26\cdots 59}{45103}$
|
| |
| Regulator: | \( 14856.117516707358 \) |
| |
| Unit signature rank: | \( 1 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{2}\cdot 14856.117516707358 \cdot 1}{2\cdot\sqrt{491371558441}}\cr\approx \mathstrut & 0.836681286178306 \end{aligned}\]
Galois group
| A solvable group of order 10 |
| The 4 conjugacy class representatives for $D_{5}$ |
| Character table for $D_{5}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Galois closure: | deg 10 |
| 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 | ${\href{/padicField/2.2.0.1}{2} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.5.0.1}{5} }$ | ${\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.1.0.1}{1} }^{5}$ | R |
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 |
|---|---|---|---|---|---|---|---|
|
\(59\)
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 59.1.2.1a1.1 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 59.1.2.1a1.1 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(109\)
| 109.1.5.4a1.1 | $x^{5} + 109$ | $5$ | $1$ | $4$ | $D_{5}$ | $$[\ ]_{5}^{2}$$ |