Basic invariants
| Dimension: | $3$ |
| Group: | $\GL(3,2)$ |
| Conductor: | \(340736\)\(\medspace = 2^{8} \cdot 11^{3} \) |
| Artin number field: | Galois closure of 7.3.1814078464.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $\PSL(2,7)$ |
| Parity: | even |
| Projective image: | $\GL(3,2)$ |
| Projective field: | Galois closure of 7.3.1814078464.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{3} + 4x + 17 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a^{2} + 18 a + 1 + \left(13 a^{2} + 12 a + 13\right)\cdot 19 + \left(17 a^{2} + 17 a + 17\right)\cdot 19^{2} + \left(3 a^{2} + 15 a + 8\right)\cdot 19^{3} + \left(4 a^{2} + 17 a + 16\right)\cdot 19^{4} + \left(9 a^{2} + 16 a + 7\right)\cdot 19^{5} + \left(5 a^{2} + 17 a + 9\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a^{2} + 9 a + 17 + \left(9 a^{2} + 5 a\right)\cdot 19 + \left(17 a^{2} + 9 a + 10\right)\cdot 19^{2} + \left(2 a^{2} + 3 a + 9\right)\cdot 19^{3} + \left(3 a^{2} + 2 a + 6\right)\cdot 19^{4} + \left(17 a^{2} + 16 a + 5\right)\cdot 19^{5} + \left(11 a^{2} + 14 a + 5\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 a^{2} + 8 a + 7 + \left(10 a^{2} + 12 a + 16\right)\cdot 19 + \left(a^{2} + 8 a + 11\right)\cdot 19^{2} + \left(17 a^{2} + 12 a + 15\right)\cdot 19^{3} + \left(9 a^{2} + 11 a + 11\right)\cdot 19^{4} + \left(9 a^{2} + a + 16\right)\cdot 19^{5} + \left(13 a^{2} + 17 a + 15\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 + 8\cdot 19 + 8\cdot 19^{2} + 18\cdot 19^{3} + 8\cdot 19^{4} +O(19^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 a^{2} + 8 a + 9 + \left(4 a^{2} + 18 a + 8\right)\cdot 19 + \left(6 a^{2} + 18 a + 12\right)\cdot 19^{2} + \left(3 a^{2} + 12 a + 13\right)\cdot 19^{3} + \left(13 a^{2} + 8\right)\cdot 19^{4} + \left(4 a^{2} + 5 a + 8\right)\cdot 19^{5} + \left(a^{2} + 15 a + 4\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a^{2} + 12 a + 2 + \left(a^{2} + 6 a + 6\right)\cdot 19 + \left(14 a^{2} + a + 14\right)\cdot 19^{2} + \left(11 a^{2} + 9 a + 10\right)\cdot 19^{3} + \left(a^{2} + 9\right)\cdot 19^{4} + \left(5 a^{2} + 16 a + 9\right)\cdot 19^{5} + \left(12 a^{2} + 4 a + 8\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 13 a^{2} + 2 a + 7 + \left(17 a^{2} + a + 3\right)\cdot 19 + \left(18 a^{2} + a + 1\right)\cdot 19^{2} + \left(17 a^{2} + 3 a + 18\right)\cdot 19^{3} + \left(5 a^{2} + 5 a + 13\right)\cdot 19^{4} + \left(11 a^{2} + a + 8\right)\cdot 19^{5} + \left(12 a^{2} + 6 a + 13\right)\cdot 19^{6} +O(19^{7})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $3$ | $3$ |
| $21$ | $2$ | $(2,6)(3,4)$ | $-1$ | $-1$ |
| $56$ | $3$ | $(1,4,5)(2,7,3)$ | $0$ | $0$ |
| $42$ | $4$ | $(1,7,4,2)(5,6)$ | $1$ | $1$ |
| $24$ | $7$ | $(1,7,3,4,6,5,2)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
| $24$ | $7$ | $(1,4,2,3,5,7,6)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |