Basic invariants
| Dimension: | $20$ |
| Group: | $S_7$ |
| Conductor: | \(202\!\cdots\!529\)\(\medspace = 23^{12} \cdot 3137^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 7.3.1659473.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 70 |
| Parity: | even |
| Projective image: | $S_7$ |
| Projective field: | Galois closure of 7.3.1659473.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$:
\( x^{2} + 82x + 3 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 65 + 64\cdot 89 + 31\cdot 89^{2} + 89^{3} + 84\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 a + 72 + 83 a\cdot 89 + \left(78 a + 66\right)\cdot 89^{2} + \left(82 a + 53\right)\cdot 89^{3} + \left(30 a + 64\right)\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 74 + 16\cdot 89 + 85\cdot 89^{2} + 75\cdot 89^{3} + 11\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 86 + 53\cdot 89 + 83\cdot 89^{2} + 63\cdot 89^{3} + 33\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 77 a + 8 + \left(19 a + 14\right)\cdot 89 + \left(24 a + 58\right)\cdot 89^{2} + \left(50 a + 39\right)\cdot 89^{3} + \left(15 a + 41\right)\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 a + 13 + \left(69 a + 76\right)\cdot 89 + \left(64 a + 29\right)\cdot 89^{2} + \left(38 a + 11\right)\cdot 89^{3} + \left(73 a + 11\right)\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 81 a + 39 + \left(5 a + 40\right)\cdot 89 + \left(10 a + 1\right)\cdot 89^{2} + \left(6 a + 21\right)\cdot 89^{3} + \left(58 a + 20\right)\cdot 89^{4} +O(89^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $20$ |
| $21$ | $2$ | $(1,2)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-4$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |