Normalized defining polynomial
\( x^{6} - x^{5} - 11709x^{4} - 133576x^{3} + 33509707x^{2} + 845596500x - 1357072575 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[6, 0]$ |
| |
| Discriminant: |
\(153517972576572213665301\)
\(\medspace = 3^{3}\cdot 17848367^{3}\)
|
| |
| Root discriminant: | \(7317.45\) |
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| Galois root discriminant: | $3^{1/2}17848367^{1/2}\approx 7317.451810569031$ | ||
| Ramified primes: |
\(3\), \(17848367\)
|
| |
| Discriminant root field: | $\Q(\sqrt{53545101}$) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{4}+\frac{1}{9}a^{3}+\frac{1}{9}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{150794646213465}a^{5}+\frac{23607408596}{16754960690385}a^{4}-\frac{878495560632}{5584986896795}a^{3}-\frac{10215377252611}{150794646213465}a^{2}-\frac{8351952416762}{16754960690385}a-\frac{1185453525587}{10052976414231}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{4}$, which has order $4$ (assuming GRH) |
| |
| Narrow class group: | $C_{2}\times C_{4}$, which has order $8$ (assuming GRH) |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{410512812875395}{150044424093}a^{5}-\frac{10\cdots 32}{50014808031}a^{4}-\frac{73\cdots 29}{50014808031}a^{3}+\frac{11\cdots 84}{150044424093}a^{2}+\frac{50\cdots 01}{16671602677}a-\frac{23\cdots 66}{50014808031}$, $\frac{32\cdots 23}{150794646213465}a^{5}+\frac{79\cdots 29}{50264882071155}a^{4}-\frac{67\cdots 29}{50264882071155}a^{3}-\frac{19\cdots 43}{150794646213465}a^{2}-\frac{40\cdots 86}{16754960690385}a+\frac{39\cdots 32}{10052976414231}$, $\frac{14\cdots 53}{150794646213465}a^{5}+\frac{41\cdots 14}{50264882071155}a^{4}-\frac{18\cdots 54}{50264882071155}a^{3}-\frac{68\cdots 68}{150794646213465}a^{2}-\frac{49\cdots 17}{5584986896795}a+\frac{14\cdots 82}{10052976414231}$, $\frac{30\cdots 24}{30158929242693}a^{5}-\frac{83\cdots 97}{10052976414231}a^{4}-\frac{53\cdots 78}{10052976414231}a^{3}+\frac{87\cdots 80}{30158929242693}a^{2}+\frac{12\cdots 51}{1116997379359}a-\frac{17\cdots 71}{10052976414231}$, $\frac{30\cdots 66}{50264882071155}a^{5}+\frac{22\cdots 59}{50264882071155}a^{4}-\frac{18\cdots 29}{50264882071155}a^{3}-\frac{59\cdots 92}{16754960690385}a^{2}-\frac{11\cdots 76}{16754960690385}a+\frac{12\cdots 23}{1116997379359}$
|
| |
| Regulator: | \( 2005544026.31 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 2005544026.31 \cdot 4}{2\cdot\sqrt{153517972576572213665301}}\cr\approx \mathstrut & 0.655182904065 \end{aligned}\] (assuming GRH)
Galois group
| A non-solvable group of order 120 |
| The 7 conjugacy class representatives for $\PGL(2,5)$ |
| Character table for $\PGL(2,5)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling algebras
| Twin sextic algebra: | \(\Q\) $\times$ 5.5.53545101.1 |
| Degree 5 sibling: | 5.5.53545101.1 |
| Degree 10 siblings: | 10.10.153517972576572213665301.1, deg 10 |
| Degree 12 sibling: | deg 12 |
| Degree 15 sibling: | deg 15 |
| Degree 20 siblings: | deg 20, deg 20, deg 20 |
| Degree 24 sibling: | deg 24 |
| Degree 30 siblings: | deg 30, deg 30, some data not computed |
| Degree 40 sibling: | deg 40 |
| Minimal sibling: | 5.5.53545101.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }$ | R | ${\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(17848367\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| Deg $4$ | $2$ | $2$ | $2$ |