Properties

Label 8.5737e4.21t14.1c1
Dimension 8
Group $\GL(3,2)$
Conductor $ 5737^{4}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$8$
Group:$\GL(3,2)$
Conductor:$1083276693622561= 5737^{4} $
Artin number field: Splitting field of $f= x^{7} - x^{6} - 2 x^{4} - 9 x^{3} + 2 x^{2} - 3 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $\PSL(2,7)$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character value
$1$$1$$()$$8$
$21$$2$$(1,3)(5,6)$$0$
$56$$3$$(1,4,6)(2,5,7)$$-1$
$42$$4$$(1,5,4,2)(3,7)$$0$
$24$$7$$(1,7,3,5,6,4,2)$$1$
$24$$7$$(1,5,2,3,4,7,6)$$1$
The blue line marks the conjugacy class containing complex conjugation.