Normalized defining polynomial
\( x^{5} - x^{4} - 99x^{3} + 602x^{2} - 472x + 212 \)
Invariants
| Degree: | $5$ |
| |
| Signature: | $(1, 2)$ |
| |
| Discriminant: |
\(85484894884\)
\(\medspace = 2^{2}\cdot 29^{2}\cdot 71^{4}\)
|
| |
| Root discriminant: | \(153.60\) |
| |
| Galois root discriminant: | $2^{2/3}29^{1/2}71^{4/5}\approx 258.7571396371764$ | ||
| Ramified primes: |
\(2\), \(29\), \(71\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{1366}a^{4}+\frac{41}{1366}a^{3}+\frac{257}{1366}a^{2}+\frac{234}{683}a+\frac{30}{683}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{5}$, which has order $5$ |
| |
| Narrow class group: | $C_{5}$, which has order $5$ |
|
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{262}{683}a^{4}+\frac{1180}{683}a^{3}-\frac{21456}{683}a^{2}+\frac{17434}{683}a-\frac{8185}{683}$, $\frac{15\cdots 18}{683}a^{4}+\frac{10\cdots 98}{683}a^{3}-\frac{92\cdots 44}{683}a^{2}+\frac{73\cdots 42}{683}a-\frac{33\cdots 09}{683}$
|
| |
| Regulator: | \( 2343.52250337 \) |
| |
| Unit signature rank: | \( 1 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{2}\cdot 2343.52250337 \cdot 5}{2\cdot\sqrt{85484894884}}\cr\approx \mathstrut & 1.58217376227 \end{aligned}\]
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: | deg 6 |
| 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 | ${\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.5.0.1}{5} }$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.5.0.1}{5} }$ | R | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }$ | ${\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.5.0.1}{5} }$ | ${\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }$ |
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 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
|
\(29\)
| $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 29.2.2.2a1.2 | $x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(71\)
| 71.1.5.4a1.4 | $x^{5} + 4118$ | $5$ | $1$ | $4$ | $C_5$ | $$[\ ]_{5}$$ |