Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(35016268618806336\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 17^{6} \cdot 23^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.2.319056.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 30T164 |
| Parity: | even |
| Projective image: | $S_6$ |
| Projective field: | Galois closure of 6.2.319056.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 401 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 401 }$:
\( x^{2} + 396x + 3 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 195 a + 103 + \left(352 a + 27\right)\cdot 401 + \left(127 a + 120\right)\cdot 401^{2} + \left(73 a + 211\right)\cdot 401^{3} + \left(331 a + 187\right)\cdot 401^{4} +O(401^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 349 + 218\cdot 401 + 191\cdot 401^{2} + 127\cdot 401^{3} + 374\cdot 401^{4} +O(401^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 206 a + 276 + \left(48 a + 391\right)\cdot 401 + \left(273 a + 5\right)\cdot 401^{2} + \left(327 a + 49\right)\cdot 401^{3} + \left(69 a + 166\right)\cdot 401^{4} +O(401^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 371 + 317\cdot 401 + 306\cdot 401^{2} + 34\cdot 401^{3} + 14\cdot 401^{4} +O(401^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 62 a + 98 + \left(377 a + 214\right)\cdot 401 + \left(139 a + 328\right)\cdot 401^{2} + \left(163 a + 51\right)\cdot 401^{3} + \left(256 a + 72\right)\cdot 401^{4} +O(401^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 339 a + 7 + \left(23 a + 33\right)\cdot 401 + \left(261 a + 250\right)\cdot 401^{2} + \left(237 a + 327\right)\cdot 401^{3} + \left(144 a + 388\right)\cdot 401^{4} +O(401^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $10$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $15$ | $2$ | $(1,2)$ | $2$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |