Properties

Label 3.15864289.42t37.a
Dimension 3
Group $\GL(3,2)$
Conductor $ 7^{2} \cdot 569^{2}$
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$3$
Group:$\GL(3,2)$
Conductor:$15864289= 7^{2} \cdot 569^{2} $
Artin number field: Splitting field of $f= x^{7} - x^{6} + x^{5} - 4 x^{4} + 4 x^{3} - 5 x^{2} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $\PSL(2,7)$
Parity: Even
Projective image: $\PSL(2,7)$
Projective field: Galois closure of 7.3.15864289.1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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