Normalized defining polynomial
\( x^{5} - 2x^{4} + 67x^{3} + 1303x^{2} + 5560x + 7207 \)
Invariants
| Degree: | $5$ |
| |
| Signature: | $(1, 2)$ |
| |
| Discriminant: |
\(102904299369\)
\(\medspace = 3^{6}\cdot 109^{4}\)
|
| |
| Root discriminant: | \(159.40\) |
| |
| Galois root discriminant: | $3^{11/6}109^{4/5}\approx 319.64174923530027$ | ||
| Ramified primes: |
\(3\), \(109\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{4}a^{3}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{48}a^{4}-\frac{1}{16}a^{3}+\frac{11}{24}a^{2}-\frac{5}{16}a+\frac{7}{48}$
| Monogenic: | No | |
| Index: | $2$ | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{11\cdots 37}{6}a^{4}-\frac{20\cdots 75}{2}a^{3}+\frac{49\cdots 43}{3}a^{2}+\frac{40\cdots 55}{2}a+\frac{25\cdots 71}{6}$, $\frac{37\cdots 53}{2}a^{4}-\frac{20\cdots 47}{2}a^{3}+16\!\cdots\!75a^{2}+\frac{38\cdots 23}{2}a+\frac{81\cdots 99}{2}$
|
| |
| Regulator: | \( 10459.368598659355 \) |
| |
| Unit signature rank: | \( 1 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{2}\cdot 10459.368598659355 \cdot 2}{2\cdot\sqrt{102904299369}}\cr\approx \mathstrut & 2.57441430865825 \end{aligned}\]
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 | ${\href{/padicField/2.2.0.1}{2} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.5.0.1}{5} }$ | ${\href{/padicField/13.5.0.1}{5} }$ | ${\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.3.5a1.3 | $x^{3} + 18 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $$[\frac{5}{2}]_{2}$$ | |
|
\(109\)
| 109.1.5.4a1.1 | $x^{5} + 109$ | $5$ | $1$ | $4$ | $D_{5}$ | $$[\ ]_{5}^{2}$$ |