Basic invariants
Dimension: | $6$ |
Group: | $\GL(3,2)$ |
Conductor: | \(36445369\)\(\medspace = 6037^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 7.3.36445369.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $\GL(3,2)$ |
Parity: | even |
Projective image: | $\GL(3,2)$ |
Projective field: | Galois closure of 7.3.36445369.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{3} + 2x + 11 \)
Roots:
$r_{ 1 }$ | $=$ | \( 2 a^{2} + 9 a + 1 + \left(7 a + 8\right)\cdot 13 + \left(a + 10\right)\cdot 13^{2} + \left(6 a^{2} + 11 a + 8\right)\cdot 13^{3} + \left(12 a^{2} + 4 a + 6\right)\cdot 13^{4} +O(13^{5})\) |
$r_{ 2 }$ | $=$ | \( 12 + 9\cdot 13 + 5\cdot 13^{2} + 6\cdot 13^{3} + 12\cdot 13^{4} +O(13^{5})\) |
$r_{ 3 }$ | $=$ | \( 6 a^{2} + 9 a + 6 + \left(2 a^{2} + 3 a + 9\right)\cdot 13 + \left(a^{2} + 5 a + 10\right)\cdot 13^{2} + \left(4 a^{2} + 11 a + 6\right)\cdot 13^{3} + \left(5 a^{2} + 10 a + 8\right)\cdot 13^{4} +O(13^{5})\) |
$r_{ 4 }$ | $=$ | \( 10 a^{2} + 5 a + 7 + \left(a^{2} + 6 a + 8\right)\cdot 13 + \left(5 a^{2} + 11\right)\cdot 13^{2} + \left(3 a^{2} + 5\right)\cdot 13^{3} + \left(11 a^{2} + 10 a + 3\right)\cdot 13^{4} +O(13^{5})\) |
$r_{ 5 }$ | $=$ | \( 3 a^{2} + 9 a + 11 + \left(10 a^{2} + 11 a + 12\right)\cdot 13 + \left(5 a^{2} + 10 a\right)\cdot 13^{2} + \left(10 a^{2} + 11 a + 6\right)\cdot 13^{3} + \left(10 a^{2} + 7 a + 4\right)\cdot 13^{4} +O(13^{5})\) |
$r_{ 6 }$ | $=$ | \( 8 a^{2} + 8 a + 9 + \left(2 a^{2} + 6 a + 2\right)\cdot 13 + \left(7 a^{2} + 7\right)\cdot 13^{2} + \left(9 a^{2} + 3 a\right)\cdot 13^{3} + \left(2 a^{2} + 11\right)\cdot 13^{4} +O(13^{5})\) |
$r_{ 7 }$ | $=$ | \( 10 a^{2} + 12 a + 7 + \left(8 a^{2} + 2 a\right)\cdot 13 + \left(6 a^{2} + 7 a + 5\right)\cdot 13^{2} + \left(5 a^{2} + a + 4\right)\cdot 13^{3} + \left(9 a^{2} + 5 a + 5\right)\cdot 13^{4} +O(13^{5})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $6$ |
$21$ | $2$ | $(2,3)(6,7)$ | $2$ |
$56$ | $3$ | $(1,5,3)(2,6,4)$ | $0$ |
$42$ | $4$ | $(1,7)(3,4,5,6)$ | $0$ |
$24$ | $7$ | $(1,7,3,2,4,5,6)$ | $-1$ |
$24$ | $7$ | $(1,2,6,3,5,7,4)$ | $-1$ |