Basic invariants
| Dimension: | $35$ |
| Group: | $S_7$ |
| Conductor: | \(193\!\cdots\!736\)\(\medspace = 2^{102} \cdot 3^{62} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 7.1.161243136.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 126 |
| Parity: | even |
| Projective image: | $S_7$ |
| Projective field: | Galois closure of 7.1.161243136.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 389 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 389 }$:
\( x^{2} + 379x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 46 + 318\cdot 389 + 188\cdot 389^{2} + 303\cdot 389^{3} + 325\cdot 389^{4} +O(389^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 226 + 81\cdot 389 + 2\cdot 389^{2} + 134\cdot 389^{3} + 3\cdot 389^{4} +O(389^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 78 a + 124 + \left(217 a + 258\right)\cdot 389 + \left(350 a + 367\right)\cdot 389^{2} + \left(324 a + 32\right)\cdot 389^{3} + \left(237 a + 251\right)\cdot 389^{4} +O(389^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 311 a + 126 + \left(171 a + 18\right)\cdot 389 + \left(38 a + 155\right)\cdot 389^{2} + \left(64 a + 208\right)\cdot 389^{3} + \left(151 a + 359\right)\cdot 389^{4} +O(389^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 339 a + 174 + \left(340 a + 227\right)\cdot 389 + \left(254 a + 363\right)\cdot 389^{2} + \left(49 a + 186\right)\cdot 389^{3} + \left(334 a + 140\right)\cdot 389^{4} +O(389^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 21 + 78\cdot 389 + 241\cdot 389^{2} + 261\cdot 389^{3} + 155\cdot 389^{4} +O(389^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 50 a + 63 + \left(48 a + 185\right)\cdot 389 + \left(134 a + 237\right)\cdot 389^{2} + \left(339 a + 39\right)\cdot 389^{3} + \left(54 a + 320\right)\cdot 389^{4} +O(389^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 7 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 7 }$ | Character values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $35$ |
| $21$ | $2$ | $(1,2)$ | $-5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |