Properties

Label 2.3204.24t22.d.a
Dimension $2$
Group $\textrm{GL(2,3)}$
Conductor $3204$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $\textrm{GL(2,3)}$
Conductor: \(3204\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 89 \)
Artin stem field: Galois closure of 8.2.24668275248.4
Galois orbit size: $2$
Smallest permutation container: 24T22
Parity: odd
Determinant: 1.267.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.9612.1

Defining polynomial

$f(x)$$=$ \( x^{8} - x^{7} + 10x^{6} - 7x^{5} + 16x^{4} - 13x^{3} - 23x^{2} + 74x + 16 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 9.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{2} + 7x + 2 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,2)(3,7)(4,8)(5,6)$$-2$
$12$$2$$(3,5)(4,8)(6,7)$$0$
$8$$3$$(1,4,3)(2,8,7)$$-1$
$6$$4$$(1,7,2,3)(4,6,8,5)$$0$
$8$$6$$(1,7,4,2,3,8)(5,6)$$1$
$6$$8$$(1,7,8,6,2,3,4,5)$$-\zeta_{8}^{3} - \zeta_{8}$
$6$$8$$(1,3,8,5,2,7,4,6)$$\zeta_{8}^{3} + \zeta_{8}$

The blue line marks the conjugacy class containing complex conjugation.