Basic invariants
| Dimension: | $8$ |
| Group: | $\GL(3,2)$ |
| Conductor: | \(18512297918464\)\(\medspace = 2^{16} \cdot 7^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.3.1475789056.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $\PSL(2,7)$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $\GL(3,2)$ |
| Projective stem field: | Galois closure of 7.3.1475789056.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 7x^{5} - 14x^{4} - 7x^{3} - 7x + 2 \)
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The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{3} + 2x + 11 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a^{2} + 6 a + 10 + \left(5 a^{2} + 10 a + 5\right)\cdot 13 + \left(7 a^{2} + 8 a + 4\right)\cdot 13^{2} + \left(5 a^{2} + 3 a + 4\right)\cdot 13^{3} + \left(2 a^{2} + a + 3\right)\cdot 13^{4} + \left(2 a^{2} + 5 a + 8\right)\cdot 13^{5} + \left(8 a^{2} + 5 a + 1\right)\cdot 13^{6} + \left(4 a + 9\right)\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 8 + \left(7 a^{2} + 7 a + 7\right)\cdot 13 + \left(10 a^{2} + 2 a + 8\right)\cdot 13^{2} + \left(4 a^{2} + 7\right)\cdot 13^{3} + 12 a\cdot 13^{4} + \left(6 a^{2} + 9\right)\cdot 13^{5} + \left(9 a^{2} + 7 a + 7\right)\cdot 13^{6} + 2 a\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 + 6\cdot 13 + 10\cdot 13^{2} + 6\cdot 13^{3} + 6\cdot 13^{5} + 3\cdot 13^{6} + 7\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 a^{2} + 5 a + 11 + \left(6 a^{2} + a + 12\right)\cdot 13 + \left(a^{2} + 6 a + 3\right)\cdot 13^{2} + \left(9 a^{2} + 12 a + 4\right)\cdot 13^{3} + \left(2 a^{2} + 7 a + 3\right)\cdot 13^{4} + \left(3 a^{2} + 10 a + 1\right)\cdot 13^{5} + \left(9 a^{2} + 3 a + 3\right)\cdot 13^{6} + \left(7 a^{2} + 8 a + 8\right)\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 a^{2} + 8 a + 2 + \left(10 a^{2} + 12 a + 1\right)\cdot 13 + \left(2 a^{2} + 9 a + 10\right)\cdot 13^{2} + \left(3 a + 9\right)\cdot 13^{3} + \left(12 a^{2} + 5 a + 2\right)\cdot 13^{4} + \left(a^{2} + 3 a + 8\right)\cdot 13^{5} + \left(10 a^{2} + 9 a + 8\right)\cdot 13^{6} + \left(11 a + 7\right)\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 5 a^{2} + 10 + \left(8 a^{2} + 12 a + 10\right)\cdot 13 + \left(8 a^{2} + 9 a + 4\right)\cdot 13^{2} + \left(3 a^{2} + 9 a + 1\right)\cdot 13^{3} + \left(11 a^{2} + 12 a + 6\right)\cdot 13^{4} + \left(7 a^{2} + 11 a + 7\right)\cdot 13^{5} + \left(6 a^{2} + 12 a + 12\right)\cdot 13^{6} + \left(4 a^{2} + 5 a + 3\right)\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 5 a^{2} + 8 a + 6 + \left(7 a + 7\right)\cdot 13 + \left(8 a^{2} + a + 9\right)\cdot 13^{2} + \left(2 a^{2} + 9 a + 4\right)\cdot 13^{3} + \left(10 a^{2} + 12 a + 9\right)\cdot 13^{4} + \left(4 a^{2} + 6 a + 11\right)\cdot 13^{5} + \left(8 a^{2} + 1\right)\cdot 13^{6} + \left(11 a^{2} + 6 a + 2\right)\cdot 13^{7} +O(13^{8})\)
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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 value |
| $1$ | $1$ | $()$ | $8$ |
| $21$ | $2$ | $(3,4)(5,6)$ | $0$ |
| $56$ | $3$ | $(2,6,5)(3,7,4)$ | $-1$ |
| $42$ | $4$ | $(1,2)(3,6,4,5)$ | $0$ |
| $24$ | $7$ | $(1,2,3,7,6,4,5)$ | $1$ |
| $24$ | $7$ | $(1,7,5,3,4,2,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.