Normalized defining polynomial
\( x^{10} - 6750x - 6075 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $[2, 4]$ |
| |
| Discriminant: |
\(76886718750000000000\)
\(\medspace = 2^{10}\cdot 3^{9}\cdot 5^{18}\)
|
| |
| Root discriminant: | \(97.41\) |
| |
| Galois root discriminant: | $2^{7/4}3^{7/6}5^{203/100}\approx 317.9457799689622$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{3}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{9}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{45}a^{5}+\frac{1}{3}a^{2}$, $\frac{1}{45}a^{6}$, $\frac{1}{135}a^{7}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{405}a^{8}-\frac{1}{135}a^{6}+\frac{1}{135}a^{5}-\frac{1}{9}a^{3}-\frac{1}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{10530}a^{9}-\frac{1}{10530}a^{8}+\frac{1}{390}a^{7}+\frac{17}{3510}a^{6}+\frac{7}{702}a^{5}-\frac{11}{234}a^{4}-\frac{5}{78}a^{3}-\frac{11}{234}a^{2}+\frac{7}{26}a+\frac{11}{26}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
| |
| Narrow class group: | Trivial group, which has order $1$ (assuming GRH) |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{585}a^{9}+\frac{17}{5265}a^{8}+\frac{16}{1755}a^{7}+\frac{2}{351}a^{6}-\frac{127}{1755}a^{5}-\frac{7}{39}a^{4}-\frac{83}{117}a^{3}-\frac{86}{117}a^{2}+\frac{20}{39}a+\frac{8}{13}$, $\frac{34}{1755}a^{9}-\frac{323}{5265}a^{8}+\frac{11}{195}a^{7}-\frac{346}{1755}a^{6}-\frac{239}{1755}a^{5}+\frac{16}{39}a^{4}-\frac{724}{117}a^{3}+\frac{1010}{117}a^{2}-\frac{1888}{39}a-\frac{724}{13}$, $\frac{11}{2106}a^{9}+\frac{3221}{10530}a^{8}+\frac{1093}{3510}a^{7}-\frac{1295}{702}a^{6}-\frac{3125}{702}a^{5}+\frac{665}{78}a^{4}+\frac{1113}{26}a^{3}-\frac{4115}{234}a^{2}-\frac{25651}{78}a-\frac{6701}{26}$, $\frac{89}{10530}a^{9}+\frac{2927}{10530}a^{8}-\frac{4789}{3510}a^{7}+\frac{5453}{1170}a^{6}-\frac{46753}{3510}a^{5}+\frac{2603}{78}a^{4}-\frac{16909}{234}a^{3}+\frac{29467}{234}a^{2}-\frac{3199}{26}a-\frac{7757}{26}$, $\frac{4729}{10530}a^{9}-\frac{753}{130}a^{8}-\frac{32639}{1170}a^{7}-\frac{53383}{702}a^{6}-\frac{26293}{234}a^{5}+\frac{19819}{234}a^{4}+\frac{284693}{234}a^{3}+\frac{117333}{26}a^{2}+\frac{790649}{78}a+\frac{164027}{26}$
|
| |
| Regulator: | \( 4374170.6999 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 4374170.6999 \cdot 1}{2\cdot\sqrt{76886718750000000000}}\cr\approx \mathstrut & 1.5549611136 \end{aligned}\] (assuming GRH)
Galois group
| A non-solvable group of order 3628800 |
| The 42 conjugacy class representatives for $S_{10}$ |
| Character table for $S_{10}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 20 sibling: | data not computed |
| Degree 45 sibling: | data not computed |
| 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 | R | R | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ |
| 2.4.2.8a5.1 | $x^{8} + 2 x^{7} + 4 x^{5} + 6 x^{4} + 2 x^{3} + 3 x^{2} + 6 x + 5$ | $2$ | $4$ | $8$ | $(C_8:C_2):C_2$ | $$[2, 2, 2]^{4}$$ | |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 3.1.3.3a1.1 | $x^{3} + 3 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $$[\frac{3}{2}]_{2}$$ | |
| 3.2.3.6a2.1 | $x^{6} + 6 x^{5} + 18 x^{4} + 32 x^{3} + 39 x^{2} + 30 x + 17$ | $3$ | $2$ | $6$ | $D_{6}$ | $$[\frac{3}{2}]_{2}^{2}$$ | |
|
\(5\)
| 5.2.5.18a1.15 | $x^{10} + 20 x^{9} + 170 x^{8} + 800 x^{7} + 2280 x^{6} + 4064 x^{5} + 4560 x^{4} + 3250 x^{3} + 1660 x^{2} + 820 x + 237$ | $5$ | $2$ | $18$ | $(C_5^2 : C_4) : C_2$ | $$[\frac{5}{4}, \frac{9}{4}]_{4}^{2}$$ |