Properties

Label 6.172764736.7t5.a.a
Dimension $6$
Group $\GL(3,2)$
Conductor $172764736$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $\GL(3,2)$
Conductor: \(172764736\)\(\medspace = 2^{6} \cdot 31^{2} \cdot 53^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 7.3.172764736.1
Galois orbit size: $1$
Smallest permutation container: $\GL(3,2)$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $\PSL(2,7)$
Projective field: Galois closure of 7.3.172764736.1

Defining polynomial

$f(x)$$=$$ x^{7} - 2 x^{6} - 2 x^{5} + 4 x^{4} - 2 x^{2} + 4 x - 2 $.

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 value
$1$$1$$()$$6$
$21$$2$$(2,6)(4,5)$$2$
$56$$3$$(1,4,5)(2,6,7)$$0$
$42$$4$$(1,7,4,2)(3,5)$$0$
$24$$7$$(1,7,4,3,5,2,6)$$-1$
$24$$7$$(1,3,6,4,2,7,5)$$-1$

The blue line marks the conjugacy class containing complex conjugation.