Basic invariants
| Dimension: | $7$ |
| Group: | $\GL(3,2)$ |
| Conductor: | \(12217597455424\)\(\medspace = 2^{6} \cdot 661^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 7.3.27962944.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $\PSL(2,7)$ |
| Parity: | even |
| Projective image: | $\PSL(2,7)$ |
| Projective field: | Galois closure of 7.3.27962944.2 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{3} + 2x + 27 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 11 a^{2} + 24 a + 18 + \left(19 a^{2} + 21 a + 2\right)\cdot 29 + \left(22 a^{2} + 24 a + 4\right)\cdot 29^{2} + \left(11 a^{2} + 28 a + 26\right)\cdot 29^{3} + \left(24 a^{2} + 15 a + 11\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 + 3\cdot 29 + 10\cdot 29^{2} + 2\cdot 29^{3} + 12\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 a^{2} + 16 a + 5 + \left(13 a^{2} + 21 a + 21\right)\cdot 29 + \left(14 a^{2} + 19 a + 22\right)\cdot 29^{2} + \left(3 a^{2} + a + 12\right)\cdot 29^{3} + \left(a^{2} + 9 a + 8\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 19 a^{2} + 12 a + 26 + \left(17 a^{2} + 8 a + 6\right)\cdot 29 + \left(2 a^{2} + 25 a + 26\right)\cdot 29^{2} + \left(16 a^{2} + 16 a + 19\right)\cdot 29^{3} + \left(13 a^{2} + 19 a + 5\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 21 a^{2} + 9 a + 12 + \left(5 a^{2} + 19 a + 13\right)\cdot 29 + \left(8 a^{2} + 5 a + 23\right)\cdot 29^{2} + \left(22 a^{2} + 20\right)\cdot 29^{3} + \left(7 a^{2} + 11 a + 18\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 26 a^{2} + 25 a + 9 + \left(3 a^{2} + 16 a + 1\right)\cdot 29 + \left(27 a^{2} + 27 a + 10\right)\cdot 29^{2} + \left(23 a^{2} + 28 a + 13\right)\cdot 29^{3} + \left(25 a^{2} + a + 23\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 14 a^{2} + a + \left(26 a^{2} + 28 a + 9\right)\cdot 29 + \left(11 a^{2} + 12 a + 19\right)\cdot 29^{2} + \left(9 a^{2} + 10 a + 20\right)\cdot 29^{3} + \left(14 a^{2} + 6\right)\cdot 29^{4} +O(29^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $7$ |
| $21$ | $2$ | $(1,7)(2,4)$ | $-1$ |
| $56$ | $3$ | $(1,5,4)(2,3,6)$ | $1$ |
| $42$ | $4$ | $(1,6,5,2)(3,7)$ | $-1$ |
| $24$ | $7$ | $(1,6,5,4,2,7,3)$ | $0$ |
| $24$ | $7$ | $(1,4,3,5,7,6,2)$ | $0$ |