Properties

Label 4.11e2_233e2.8t23.1c1
Dimension 4
Group $\textrm{GL(2,3)}$
Conductor $ 11^{2} \cdot 233^{2}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$\textrm{GL(2,3)}$
Conductor:$6568969= 11^{2} \cdot 233^{2} $
Artin number field: Splitting field of $f= x^{8} - 4 x^{6} - 4 x^{5} + 13 x^{4} + x^{3} - 60 x^{2} - 75 x - 41 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $\textrm{GL(2,3)}$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

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

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation
$(1,8,7,2)(3,6,5,4)$
$(1,2,4)(6,7,8)$
$(1,7)(2,8)(3,5)(4,6)$
$(1,6,7,4)(2,5,8,3)$
$(1,7)(2,6)(4,8)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$4$
$1$$2$$(1,7)(2,8)(3,5)(4,6)$$-4$
$12$$2$$(1,7)(2,6)(4,8)$$0$
$8$$3$$(1,2,4)(6,7,8)$$1$
$6$$4$$(1,6,7,4)(2,5,8,3)$$0$
$8$$6$$(1,5,8,7,3,2)(4,6)$$-1$
$6$$8$$(1,4,3,2,7,6,5,8)$$0$
$6$$8$$(1,6,3,8,7,4,5,2)$$0$
The blue line marks the conjugacy class containing complex conjugation.