Properties

Label 3.12241352.42t37.a
Dimension $3$
Group $\GL(3,2)$
Conductor $12241352$
Indicator $0$

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Basic invariants

Dimension:$3$
Group:$\GL(3,2)$
Conductor:\(12241352\)\(\medspace = 2^{3} \cdot 1237^{2} \)
Artin number field: Galois closure of 7.3.97930816.2
Galois orbit size: $2$
Smallest permutation container: $\PSL(2,7)$
Parity: even
Projective image: $\GL(3,2)$
Projective field: Galois closure of 7.3.97930816.2

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{3} + x + 14 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 5 a^{2} + 4 a + 1 + \left(9 a^{2} + 15 a\right)\cdot 17 + \left(2 a^{2} + 6 a + 16\right)\cdot 17^{2} + \left(6 a^{2} + 14 a + 4\right)\cdot 17^{3} + \left(11 a^{2} + 12 a + 11\right)\cdot 17^{4} + \left(7 a^{2} + a + 13\right)\cdot 17^{5} + \left(10 a^{2} + 15 a + 5\right)\cdot 17^{6} +O(17^{7})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 12 a^{2} + 10 a + \left(10 a^{2} + a + 1\right)\cdot 17 + \left(2 a^{2} + 11 a + 16\right)\cdot 17^{2} + \left(8 a^{2} + 7 a + 11\right)\cdot 17^{3} + \left(3 a^{2} + 11\right)\cdot 17^{4} + \left(16 a^{2} + 16 a + 13\right)\cdot 17^{5} + \left(5 a^{2} + 3 a + 2\right)\cdot 17^{6} +O(17^{7})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 16 + 15\cdot 17 + 12\cdot 17^{2} + 5\cdot 17^{3} + 16\cdot 17^{4} + 9\cdot 17^{5} + 14\cdot 17^{6} +O(17^{7})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( a + 14 + \left(a^{2} + 14 a + 12\right)\cdot 17 + \left(7 a^{2} + 5 a + 8\right)\cdot 17^{2} + \left(4 a^{2} + a + 11\right)\cdot 17^{3} + \left(2 a^{2} + 4 a + 3\right)\cdot 17^{4} + \left(7 a^{2} + 16 a + 4\right)\cdot 17^{5} + \left(10 a + 2\right)\cdot 17^{6} +O(17^{7})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 13 a^{2} + 7 a + \left(4 a^{2} + 15 a + 4\right)\cdot 17 + \left(15 a^{2} + 14 a + 14\right)\cdot 17^{2} + \left(a^{2} + 4 a + 9\right)\cdot 17^{3} + \left(11 a^{2} + 8 a + 9\right)\cdot 17^{4} + \left(4 a^{2} + 7 a + 2\right)\cdot 17^{5} + \left(7 a^{2} + 13 a + 1\right)\cdot 17^{6} +O(17^{7})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 4 a^{2} + 9 a + 11 + \left(11 a^{2} + 4 a + 2\right)\cdot 17 + \left(11 a^{2} + 13 a + 6\right)\cdot 17^{2} + \left(10 a^{2} + 10 a + 4\right)\cdot 17^{3} + \left(3 a^{2} + 4 a + 10\right)\cdot 17^{4} + \left(5 a^{2} + 10 a + 8\right)\cdot 17^{5} + \left(9 a^{2} + 9 a + 2\right)\cdot 17^{6} +O(17^{7})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 3 a + 9 + \left(14 a^{2} + 14\right)\cdot 17 + \left(11 a^{2} + 16 a + 10\right)\cdot 17^{2} + \left(2 a^{2} + 11 a + 2\right)\cdot 17^{3} + \left(2 a^{2} + 3 a + 5\right)\cdot 17^{4} + \left(10 a^{2} + 16 a + 15\right)\cdot 17^{5} + \left(14 a + 4\right)\cdot 17^{6} +O(17^{7})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

Cycle notation
$(1,6)(5,7)$
$(2,5)(3,4,6,7)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $3$ $3$
$21$ $2$ $(1,6)(5,7)$ $-1$ $-1$
$56$ $3$ $(1,7,4)(2,3,6)$ $0$ $0$
$42$ $4$ $(2,5)(3,4,6,7)$ $1$ $1$
$24$ $7$ $(1,7,2,5,3,4,6)$ $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$
$24$ $7$ $(1,5,6,2,4,7,3)$ $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$
The blue line marks the conjugacy class containing complex conjugation.