Basic invariants
| Dimension: | $7$ |
| Group: | $\PGL(2,7)$ |
| Conductor: | \(1340095640625\)\(\medspace = 3^{6} \cdot 5^{6} \cdot 7^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.9380669484375.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T713 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $\PGL(2,7)$ |
| Projective stem field: | Galois closure of 8.2.9380669484375.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 7x^{6} - 28x^{5} + 70x^{4} - 112x^{3} + 112x^{2} - 64x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{3} + 6x + 35 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 33 a^{2} + 1 + \left(9 a^{2} + 3 a + 22\right)\cdot 37 + \left(21 a^{2} + 36 a + 1\right)\cdot 37^{2} + \left(9 a^{2} + 11 a + 2\right)\cdot 37^{3} + \left(21 a^{2} + 27 a + 1\right)\cdot 37^{4} + \left(4 a^{2} + 27 a + 27\right)\cdot 37^{5} + \left(36 a^{2} + 10 a + 32\right)\cdot 37^{6} + \left(9 a^{2} + 26 a + 14\right)\cdot 37^{7} + \left(24 a^{2} + 8 a + 9\right)\cdot 37^{8} + \left(11 a + 9\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 a^{2} + 11 a + 2 + \left(18 a^{2} + 36 a + 7\right)\cdot 37 + \left(12 a^{2} + a + 16\right)\cdot 37^{2} + \left(9 a^{2} + 3 a + 30\right)\cdot 37^{3} + \left(4 a^{2} + 21 a + 19\right)\cdot 37^{4} + \left(12 a^{2} + 2 a + 18\right)\cdot 37^{5} + \left(35 a^{2} + 26 a + 34\right)\cdot 37^{6} + \left(a^{2} + 8 a + 18\right)\cdot 37^{7} + \left(29 a^{2} + 27 a + 11\right)\cdot 37^{8} + \left(7 a^{2} + 5 a + 7\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 + 3\cdot 37 + 29\cdot 37^{2} + 25\cdot 37^{3} + 27\cdot 37^{4} + 7\cdot 37^{5} + 2\cdot 37^{6} + 20\cdot 37^{7} + 30\cdot 37^{8} + 15\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 a^{2} + 14 a + 15 + \left(21 a^{2} + 13 a + 18\right)\cdot 37 + \left(25 a^{2} + 7 a + 31\right)\cdot 37^{2} + \left(2 a^{2} + 5 a + 3\right)\cdot 37^{3} + \left(29 a^{2} + 23 a + 8\right)\cdot 37^{4} + \left(2 a^{2} + 18\right)\cdot 37^{5} + \left(33 a^{2} + 11 a + 25\right)\cdot 37^{6} + \left(27 a^{2} + 31 a + 11\right)\cdot 37^{7} + \left(33 a^{2} + 32 a + 30\right)\cdot 37^{8} + \left(15 a + 16\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 a^{2} + 18 a + 7 + \left(11 a^{2} + 36 a + 28\right)\cdot 37 + \left(35 a^{2} + 8 a + 20\right)\cdot 37^{2} + \left(20 a^{2} + 9 a + 10\right)\cdot 37^{3} + \left(21 a^{2} + 10 a + 2\right)\cdot 37^{4} + \left(8 a^{2} + 16 a + 6\right)\cdot 37^{5} + \left(14 a^{2} + 22 a + 19\right)\cdot 37^{6} + \left(12 a^{2} + 29 a + 24\right)\cdot 37^{7} + \left(26 a^{2} + 30 a + 17\right)\cdot 37^{8} + \left(14 a^{2} + 14 a + 28\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 + 27\cdot 37 + 26\cdot 37^{2} + 29\cdot 37^{3} + 30\cdot 37^{4} + 18\cdot 37^{5} + 24\cdot 37^{6} + 21\cdot 37^{7} + 28\cdot 37^{8} + 35\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 25 a^{2} + 19 a + 6 + \left(15 a^{2} + 34 a + 8\right)\cdot 37 + \left(17 a^{2} + 28 a + 23\right)\cdot 37^{2} + \left(6 a^{2} + 15 a + 26\right)\cdot 37^{3} + \left(31 a^{2} + 36 a + 3\right)\cdot 37^{4} + \left(23 a^{2} + 29 a + 30\right)\cdot 37^{5} + \left(23 a^{2} + 3 a + 19\right)\cdot 37^{6} + \left(14 a^{2} + 18 a + 33\right)\cdot 37^{7} + \left(23 a^{2} + 34 a + 5\right)\cdot 37^{8} + \left(21 a^{2} + 10 a + 19\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 9 a^{2} + 12 a + 7 + \left(34 a^{2} + 24 a + 33\right)\cdot 37 + \left(35 a^{2} + 27 a + 35\right)\cdot 37^{2} + \left(24 a^{2} + 28 a + 18\right)\cdot 37^{3} + \left(3 a^{2} + 29 a + 17\right)\cdot 37^{4} + \left(22 a^{2} + 33 a + 21\right)\cdot 37^{5} + \left(5 a^{2} + 36 a + 26\right)\cdot 37^{6} + \left(7 a^{2} + 33 a + 2\right)\cdot 37^{7} + \left(11 a^{2} + 13 a + 14\right)\cdot 37^{8} + \left(28 a^{2} + 15 a + 15\right)\cdot 37^{9} +O(37^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $7$ |
| $21$ | $2$ | $(1,7)(2,5)(3,4)(6,8)$ | $-1$ |
| $28$ | $2$ | $(1,5)(3,8)(4,7)$ | $-1$ |
| $56$ | $3$ | $(1,8,7)(3,4,5)$ | $1$ |
| $42$ | $4$ | $(1,7,6,2)(3,4,8,5)$ | $-1$ |
| $56$ | $6$ | $(1,4,8,5,7,3)$ | $-1$ |
| $48$ | $7$ | $(2,3,4,6,8,5,7)$ | $0$ |
| $42$ | $8$ | $(1,3,7,4,6,8,2,5)$ | $1$ |
| $42$ | $8$ | $(1,4,2,3,6,5,7,8)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.