Basic invariants
| Dimension: | $7$ |
| Group: | $\PGL(2,7)$ |
| Conductor: | \(345808999872\)\(\medspace = 2^{6} \cdot 3^{8} \cdot 7^{7} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.345808999872.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $\PGL(2,7)$ |
| Parity: | odd |
| Determinant: | 1.7.2t1.a.a |
| Projective image: | $\PGL(2,7)$ |
| Projective stem field: | Galois closure of 8.2.345808999872.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 14x^{5} - 84x^{3} + 112x^{2} - 24x - 12 \)
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The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{3} + 2x + 27 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 3\cdot 29 + 21\cdot 29^{2} + 29^{3} + 28\cdot 29^{4} + 11\cdot 29^{5} + 11\cdot 29^{6} + 28\cdot 29^{7} + 6\cdot 29^{8} + 20\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 a^{2} + 25 a + 20 + \left(8 a^{2} + 19 a + 16\right)\cdot 29 + \left(26 a^{2} + 23 a + 16\right)\cdot 29^{2} + \left(28 a^{2} + 16 a + 17\right)\cdot 29^{3} + \left(5 a^{2} + 22 a + 24\right)\cdot 29^{4} + \left(8 a^{2} + 24 a + 6\right)\cdot 29^{5} + \left(10 a^{2} + 7 a + 21\right)\cdot 29^{6} + \left(a^{2} + 16 a + 27\right)\cdot 29^{7} + \left(4 a^{2} + 27 a + 3\right)\cdot 29^{8} + \left(14 a^{2} + 4 a + 8\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 a^{2} + 24 a + 17 + \left(13 a^{2} + 4 a + 23\right)\cdot 29 + \left(8 a^{2} + 3 a + 21\right)\cdot 29^{2} + \left(9 a^{2} + 12 a + 10\right)\cdot 29^{3} + \left(18 a^{2} + 11 a + 2\right)\cdot 29^{4} + \left(17 a^{2} + 23 a\right)\cdot 29^{5} + \left(20 a^{2} + 26 a + 6\right)\cdot 29^{6} + \left(7 a^{2} + 3 a + 7\right)\cdot 29^{7} + \left(25 a^{2} + 7 a + 3\right)\cdot 29^{8} + \left(24 a^{2} + a + 3\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 14 a^{2} + 25 a + 9 + \left(25 a^{2} + a + 8\right)\cdot 29 + \left(25 a^{2} + 26 a + 2\right)\cdot 29^{2} + \left(4 a^{2} + 10 a + 5\right)\cdot 29^{3} + \left(22 a^{2} + 3 a + 10\right)\cdot 29^{4} + \left(22 a^{2} + 14 a + 1\right)\cdot 29^{5} + \left(12 a^{2} + 23 a + 28\right)\cdot 29^{6} + \left(6 a^{2} + 7 a + 22\right)\cdot 29^{7} + \left(18 a^{2} + 13 a + 6\right)\cdot 29^{8} + \left(17 a^{2} + 9 a + 23\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 a^{2} + 9 a + 26 + \left(6 a^{2} + 4 a + 3\right)\cdot 29 + \left(23 a^{2} + 2 a + 22\right)\cdot 29^{2} + \left(19 a^{2} + 24\right)\cdot 29^{3} + \left(4 a^{2} + 24 a + 22\right)\cdot 29^{4} + \left(3 a^{2} + 9 a + 9\right)\cdot 29^{5} + \left(27 a^{2} + 23 a + 24\right)\cdot 29^{6} + \left(19 a^{2} + 8 a + 13\right)\cdot 29^{7} + \left(28 a^{2} + 23 a + 17\right)\cdot 29^{8} + \left(18 a^{2} + 22 a + 14\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 a^{2} + 17 a + 5 + \left(a^{2} + 14 a + 5\right)\cdot 29 + \left(2 a^{2} + 20 a + 9\right)\cdot 29^{2} + \left(5 a^{2} + 4 a + 5\right)\cdot 29^{3} + \left(a^{2} + 4 a + 11\right)\cdot 29^{4} + \left(17 a^{2} + 20 a + 3\right)\cdot 29^{5} + \left(11 a^{2} + 7 a + 7\right)\cdot 29^{6} + \left(5 a^{2} + 13 a + 2\right)\cdot 29^{7} + \left(2 a^{2} + 26 a + 24\right)\cdot 29^{8} + \left(27 a^{2} + 8 a + 6\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 23 + 15\cdot 29^{2} + 7\cdot 29^{3} + 9\cdot 29^{4} + 20\cdot 29^{6} + 24\cdot 29^{7} + 20\cdot 29^{8} + 12\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 4 a^{2} + 16 a + 15 + \left(2 a^{2} + 12 a + 25\right)\cdot 29 + \left(a^{2} + 11 a + 7\right)\cdot 29^{2} + \left(19 a^{2} + 13 a + 14\right)\cdot 29^{3} + \left(5 a^{2} + 21 a + 7\right)\cdot 29^{4} + \left(18 a^{2} + 23 a + 24\right)\cdot 29^{5} + \left(4 a^{2} + 26 a + 26\right)\cdot 29^{6} + \left(17 a^{2} + 7 a + 17\right)\cdot 29^{7} + \left(8 a^{2} + 18 a + 3\right)\cdot 29^{8} + \left(13 a^{2} + 10 a + 27\right)\cdot 29^{9} +O(29^{10})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
| $1$ | $1$ | $()$ | $7$ |
| $21$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $-1$ |
| $28$ | $2$ | $(1,7)(2,5)(4,8)$ | $1$ |
| $56$ | $3$ | $(1,4,5)(2,7,8)$ | $1$ |
| $42$ | $4$ | $(1,4,5,8)(2,6,3,7)$ | $-1$ |
| $56$ | $6$ | $(1,2,4,7,5,8)$ | $1$ |
| $48$ | $7$ | $(1,2,3,4,5,6,8)$ | $0$ |
| $42$ | $8$ | $(1,7,4,2,5,6,8,3)$ | $-1$ |
| $42$ | $8$ | $(1,2,8,7,5,3,4,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.