Normalized defining polynomial
\( x^{5} - x^{4} - 65x^{3} + 13x^{2} + 1274x - 548 \)
Invariants
| Degree: | $5$ |
| |
| Signature: | $(1, 2)$ |
| |
| Discriminant: |
\(411617197476\)
\(\medspace = 2^{2}\cdot 3^{6}\cdot 109^{4}\)
|
| |
| Root discriminant: | \(210.33\) |
| |
| Galois root discriminant: | $2^{2/3}3^{11/6}109^{4/5}\approx 507.39964898907107$ | ||
| Ramified primes: |
\(2\), \(3\), \(109\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{6}a^{4}-\frac{1}{6}a^{3}-\frac{1}{2}a^{2}+\frac{1}{6}a+\frac{1}{3}$
| 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: |
$\frac{9793108408}{3}a^{4}+16647655123a^{3}-110696125372a^{2}-\frac{1900325741273}{3}a+293117882411$, $\frac{19\cdots 71}{2}a^{4}+\frac{98\cdots 43}{2}a^{3}-\frac{65\cdots 21}{2}a^{2}-\frac{37\cdots 91}{2}a+86\!\cdots\!73$
|
| |
| Regulator: | \( 25746.54406 \) |
| |
| Unit signature rank: | \( 1 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{2}\cdot 25746.54406 \cdot 1}{2\cdot\sqrt{411617197476}}\cr\approx \mathstrut & 1.58427994 \end{aligned}\]
Galois group
| A non-solvable group of order 60 |
| The 5 conjugacy class representatives for $A_5$ |
| Character table for $A_5$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 6 sibling: | 6.2.133363971982224.1 |
| Degree 10 sibling: | deg 10 |
| Degree 12 sibling: | deg 12 |
| Degree 15 sibling: | deg 15 |
| Degree 20 sibling: | deg 20 |
| Degree 30 sibling: | deg 30 |
| 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 | ${\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.1.0.1}{1} }^{5}$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }$ | ${\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 2.1.3.2a1.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
|
\(3\)
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.3.5a1.3 | $x^{3} + 18 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $$[\frac{5}{2}]_{2}$$ | |
|
\(109\)
| 109.1.5.4a1.1 | $x^{5} + 109$ | $5$ | $1$ | $4$ | $D_{5}$ | $$[\ ]_{5}^{2}$$ |