Normalized defining polynomial
\( x^{5} - 3x^{3} - 3x^{2} + 9x - 9 \)
Invariants
| Degree: | $5$ |
| |
| Signature: | $[1, 2]$ |
| |
| Discriminant: |
\(363609\)
\(\medspace = 3^{4}\cdot 67^{2}\)
|
| |
| Root discriminant: | \(12.95\) |
| |
| Galois root discriminant: | $3^{7/6}67^{1/2}\approx 29.49027790468186$ | ||
| Ramified primes: |
\(3\), \(67\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{3}a^{4}$
| 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: |
$\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-a^{2}-2a+1$, $\frac{1}{3}a^{4}-a^{3}-3a^{2}+11$
|
| |
| Regulator: | \( 11.7316074152 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{2}\cdot 11.7316074152 \cdot 1}{2\cdot\sqrt{363609}}\cr\approx \mathstrut & 0.768068485419 \end{aligned}\]
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: | 6.2.3272481.2 |
| Degree 10 sibling: | 10.2.1189903543929.1 |
| 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 | ${\href{/padicField/2.5.0.1}{5} }$ | R | ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.5.0.1}{5} }$ | ${\href{/padicField/13.5.0.1}{5} }$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }$ | ${\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.3.3a1.1 | $x^{3} + 3 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $$[\frac{3}{2}]_{2}$$ | |
|
\(67\)
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 67.2.2.2a1.2 | $x^{4} + 126 x^{3} + 3973 x^{2} + 252 x + 71$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 3.363609.12t33.a.a | $3$ | $ 3^{4} \cdot 67^{2}$ | 5.1.363609.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
| 3.363609.12t33.a.b | $3$ | $ 3^{4} \cdot 67^{2}$ | 5.1.363609.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
| * | 4.363609.5t4.a.a | $4$ | $ 3^{4} \cdot 67^{2}$ | 5.1.363609.1 | $A_5$ (as 5T4) | $1$ | $0$ |
| 5.3272481.6t12.a.a | $5$ | $ 3^{6} \cdot 67^{2}$ | 5.1.363609.1 | $A_5$ (as 5T4) | $1$ | $1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.