Basic invariants
| Dimension: | $35$ |
| Group: | $S_7$ |
| Conductor: | \(253\!\cdots\!375\)\(\medspace = 5^{24} \cdot 37^{15} \cdot 347^{15} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 7.1.320975.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 70 |
| Parity: | odd |
| Projective image: | $S_7$ |
| Projective field: | Galois closure of 7.1.320975.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 44 a + 35 + \left(25 a + 4\right)\cdot 53 + \left(49 a + 10\right)\cdot 53^{2} + \left(11 a + 34\right)\cdot 53^{3} + 35\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 + 21\cdot 53 + 16\cdot 53^{2} + 29\cdot 53^{3} + 35\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a + 52 + \left(27 a + 10\right)\cdot 53 + \left(3 a + 23\right)\cdot 53^{2} + \left(41 a + 32\right)\cdot 53^{3} + \left(52 a + 24\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 52 a + 51 + \left(41 a + 51\right)\cdot 53 + \left(20 a + 39\right)\cdot 53^{2} + \left(31 a + 47\right)\cdot 53^{3} + \left(25 a + 16\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 31 + 2\cdot 53 + 52\cdot 53^{2} + 10\cdot 53^{3} + 26\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( a + 47 + \left(11 a + 8\right)\cdot 53 + \left(32 a + 28\right)\cdot 53^{2} + \left(21 a + 46\right)\cdot 53^{3} + \left(27 a + 34\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 36 + 5\cdot 53 + 42\cdot 53^{2} + 10\cdot 53^{3} + 38\cdot 53^{4} +O(53^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $35$ |
| $21$ | $2$ | $(1,2)$ | $5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |