Basic invariants
| Dimension: | $35$ |
| Group: | $S_7$ |
| Conductor: | \(121\!\cdots\!201\)\(\medspace = 17^{20} \cdot 23^{20} \cdot 64879^{20} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.7.25367689.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 126 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.7.25367689.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} - 6x^{5} + 4x^{4} + 9x^{3} - 4x^{2} - 3x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 353 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 353 }$:
\( x^{2} + 348x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 306 + 81\cdot 353 + 153\cdot 353^{2} + 289\cdot 353^{3} + 13\cdot 353^{4} +O(353^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 124 a + 13 + \left(4 a + 102\right)\cdot 353 + \left(266 a + 141\right)\cdot 353^{2} + \left(139 a + 292\right)\cdot 353^{3} + \left(10 a + 341\right)\cdot 353^{4} +O(353^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 318 a + 298 + \left(37 a + 330\right)\cdot 353 + 301\cdot 353^{2} + \left(189 a + 227\right)\cdot 353^{3} + \left(22 a + 271\right)\cdot 353^{4} +O(353^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 86 + 238\cdot 353 + 103\cdot 353^{2} + 195\cdot 353^{3} + 26\cdot 353^{4} +O(353^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 229 a + 280 + \left(348 a + 352\right)\cdot 353 + \left(86 a + 54\right)\cdot 353^{2} + \left(213 a + 19\right)\cdot 353^{3} + \left(342 a + 254\right)\cdot 353^{4} +O(353^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 307 + 103\cdot 353 + 39\cdot 353^{2} + 274\cdot 353^{3} + 308\cdot 353^{4} +O(353^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 35 a + 123 + \left(315 a + 202\right)\cdot 353 + \left(352 a + 264\right)\cdot 353^{2} + \left(163 a + 113\right)\cdot 353^{3} + \left(330 a + 195\right)\cdot 353^{4} +O(353^{5})\)
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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 value |
| $1$ | $1$ | $()$ | $35$ |
| $21$ | $2$ | $(1,2)$ | $-5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.