Normalized defining polynomial
\( x^{5} - x^{4} + 44x^{3} + 122x^{2} + 184x - 112 \)
Invariants
| Degree: | $5$ |
| |
| Signature: | $(1, 2)$ |
| |
| Discriminant: |
\(652715336464\)
\(\medspace = 2^{4}\cdot 17^{2}\cdot 109^{4}\)
|
| |
| Root discriminant: | \(230.65\) |
| |
| Galois root discriminant: | $2\cdot 17^{1/2}109^{4/5}\approx 351.71885826969174$ | ||
| Ramified primes: |
\(2\), \(17\), \(109\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{28}a^{4}-\frac{3}{28}a^{3}-\frac{3}{14}a^{2}-\frac{3}{14}a$
| Monogenic: | No | |
| Index: | $4$ | |
| 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: |
$\frac{55\cdots 53}{28}a^{4}-\frac{74\cdots 93}{28}a^{3}-\frac{56\cdots 03}{7}a^{2}-\frac{16\cdots 35}{14}a+71\!\cdots\!99$, $\frac{23\cdots 33}{7}a^{4}+\frac{64\cdots 58}{7}a^{3}+\frac{84\cdots 88}{7}a^{2}-\frac{46\cdots 53}{7}a-46\!\cdots\!01$
|
| |
| Regulator: | \( 143805.836682 \) |
| |
| Unit signature rank: | \( 1 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{2}\cdot 143805.836682 \cdot 1}{2\cdot\sqrt{652715336464}}\cr\approx \mathstrut & 7.0270709963 \end{aligned}\]
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 | R | ${\href{/padicField/3.5.0.1}{5} }$ | ${\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.1.0.1}{1} }^{5}$ | ${\href{/padicField/11.1.0.1}{1} }^{5}$ | ${\href{/padicField/13.5.0.1}{5} }$ | R | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.5.0.1}{5} }$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ | |
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ | |
|
\(17\)
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 17.1.2.1a1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 17.1.2.1a1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(109\)
| 109.1.5.4a1.1 | $x^{5} + 109$ | $5$ | $1$ | $4$ | $D_{5}$ | $$[\ ]_{5}^{2}$$ |