Properties

Label 3.3e4_491e2.42t37.1
Dimension 3
Group $\GL(3,2)$
Conductor $ 3^{4} \cdot 491^{2}$
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$3$
Group:$\GL(3,2)$
Conductor:$19527561= 3^{4} \cdot 491^{2} $
Artin number field: Splitting field of $f= x^{7} - 2 x^{6} + 3 x^{4} - 3 x^{3} + 3 x^{2} - 4 x - 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $\PSL(2,7)$
Parity: Even

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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