Properties

Label 3.2e3_31e2_53e2.42t37.1
Dimension 3
Group $\GL(3,2)$
Conductor $ 2^{3} \cdot 31^{2} \cdot 53^{2}$
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$3$
Group:$\GL(3,2)$
Conductor:$21595592= 2^{3} \cdot 31^{2} \cdot 53^{2} $
Artin number field: Splitting field of $f= x^{7} - 2 x^{6} - 2 x^{5} + 4 x^{4} - 2 x^{2} + 4 x - 2 $ 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:
$r_{ 1 }$ $=$ $ 2 a^{2} + 7 a + 9 + \left(2 a^{2} + 8 a + 7\right)\cdot 11 + \left(10 a^{2} + a + 5\right)\cdot 11^{2} + \left(9 a^{2} + 8 a + 6\right)\cdot 11^{3} + \left(6 a^{2} + 5 a + 1\right)\cdot 11^{4} + \left(9 a^{2} + 9 a + 9\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 8 a^{2} + a + \left(3 a^{2} + 3 a + 6\right)\cdot 11 + \left(7 a^{2} + 10 a + 2\right)\cdot 11^{2} + \left(9 a^{2} + 3 a + 4\right)\cdot 11^{3} + \left(7 a^{2} + a + 9\right)\cdot 11^{4} + \left(8 a^{2} + 9 a + 10\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 4 + 4\cdot 11 + 11^{2} + 2\cdot 11^{3} + 4\cdot 11^{4} + 2\cdot 11^{5} +O\left(11^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 6 a^{2} + 3 a + 7 + \left(7 a^{2} + 3 a + 7\right)\cdot 11 + 7\cdot 11^{2} + \left(9 a^{2} + 6 a + 1\right)\cdot 11^{3} + \left(3 a^{2} + a + 1\right)\cdot 11^{4} + \left(9 a^{2} + 5 a + 5\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 7 a^{2} + 6 + \left(5 a^{2} + 8\right)\cdot 11 + \left(8 a^{2} + 2 a + 7\right)\cdot 11^{2} + \left(8 a^{2} + 5 a + 6\right)\cdot 11^{3} + \left(2 a^{2} + 3 a + 2\right)\cdot 11^{4} + \left(9 a^{2} + 6 a + 4\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 3 a^{2} + a + 3 + \left(a^{2} + 10 a + 10\right)\cdot 11 + \left(8 a + 6\right)\cdot 11^{2} + \left(3 a^{2} + 7 a + 4\right)\cdot 11^{3} + \left(3 a + 7\right)\cdot 11^{4} + \left(3 a^{2} + 7 a + 7\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 7 a^{2} + 10 a + 6 + \left(a^{2} + 7 a + 10\right)\cdot 11 + \left(6 a^{2} + 9 a\right)\cdot 11^{2} + \left(3 a^{2} + a + 7\right)\cdot 11^{3} + \left(6 a + 6\right)\cdot 11^{4} + \left(4 a^{2} + 6 a + 4\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$

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

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

Character values on conjugacy classes

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