Properties

Label 8.646274503744.21t14.a.a
Dimension $8$
Group $\GL(3,2)$
Conductor $646274503744$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $8$
Group: $\GL(3,2)$
Conductor: \(646274503744\)\(\medspace = 2^{6} \cdot 317^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.3.6431296.1
Galois orbit size: $1$
Smallest permutation container: $\PSL(2,7)$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $\GL(3,2)$
Projective stem field: Galois closure of 7.3.6431296.1

Defining polynomial

$f(x)$$=$ \( x^{7} - 2x^{6} + 2x^{4} - 2x^{3} + 2x^{2} - 2 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{3} + 2x + 9 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 2 + 10\cdot 11 + 10\cdot 11^{2} + 6\cdot 11^{3} + 8\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 6 a^{2} + 7 a + 6 + \left(a^{2} + 8\right)\cdot 11 + \left(8 a^{2} + 10 a + 2\right)\cdot 11^{2} + \left(6 a^{2} + 10 a + 3\right)\cdot 11^{3} + \left(3 a^{2} + 4 a + 7\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 3 a^{2} + a + 6 + \left(9 a^{2} + 3 a + 2\right)\cdot 11 + \left(6 a^{2} + 10 a + 6\right)\cdot 11^{2} + \left(7 a^{2} + 5 a + 2\right)\cdot 11^{3} + \left(4 a^{2} + 3 a + 8\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 10 a^{2} + 4 a + 4 + \left(7 a^{2} + 4 a + 2\right)\cdot 11 + \left(8 a^{2} + 7\right)\cdot 11^{2} + \left(6 a^{2} + a + 10\right)\cdot 11^{3} + \left(4 a^{2} + a + 4\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 6 a^{2} + 6 + \left(a^{2} + 6 a + 8\right)\cdot 11 + \left(5 a^{2} + 9\right)\cdot 11^{2} + \left(8 a^{2} + 10 a + 1\right)\cdot 11^{3} + \left(2 a^{2} + 4 a + 6\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 2 a^{2} + 7 a + 1 + \left(6 a^{2} + 2 a + 2\right)\cdot 11 + 5\cdot 11^{2} + \left(7 a^{2} + 5\right)\cdot 11^{3} + \left(5 a^{2} + 7 a + 9\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 6 a^{2} + 3 a + 10 + \left(6 a^{2} + 5 a + 9\right)\cdot 11 + \left(3 a^{2} + 1\right)\cdot 11^{2} + \left(7 a^{2} + 5 a + 2\right)\cdot 11^{3} + 10\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character value
$1$$1$$()$$8$
$21$$2$$(1,2)(3,6)$$0$
$56$$3$$(1,2,5)(3,6,7)$$-1$
$42$$4$$(1,3,5,7)(2,4)$$0$
$24$$7$$(1,4,2,3,6,5,7)$$1$
$24$$7$$(1,3,7,2,5,4,6)$$1$

The blue line marks the conjugacy class containing complex conjugation.