Properties

Label 4.272484.8t23.b.a
Dimension $4$
Group $\textrm{GL(2,3)}$
Conductor $272484$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $\textrm{GL(2,3)}$
Conductor: \(272484\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 29^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.2.24749176752.1
Galois orbit size: $1$
Smallest permutation container: $\textrm{GL(2,3)}$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.90828.2

Defining polynomial

$f(x)$$=$ \( x^{8} - 4x^{7} + 7x^{6} - 7x^{5} + 7x^{4} - 7x^{3} - 26x^{2} + 29x - 8 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 6 a + 6 + \left(15 a + 22\right)\cdot 23 + \left(22 a + 7\right)\cdot 23^{2} + \left(20 a + 13\right)\cdot 23^{3} + \left(7 a + 2\right)\cdot 23^{4} + \left(8 a + 7\right)\cdot 23^{5} + \left(10 a + 5\right)\cdot 23^{6} + \left(3 a + 13\right)\cdot 23^{7} + \left(2 a + 22\right)\cdot 23^{8} + \left(8 a + 15\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 4 + 11\cdot 23 + 10\cdot 23^{2} + 9\cdot 23^{4} + 4\cdot 23^{5} + 16\cdot 23^{6} + 13\cdot 23^{8} + 18\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 20 + 11\cdot 23 + 12\cdot 23^{2} + 22\cdot 23^{3} + 13\cdot 23^{4} + 18\cdot 23^{5} + 6\cdot 23^{6} + 22\cdot 23^{7} + 9\cdot 23^{8} + 4\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 12 a + 15 + \left(11 a + 17\right)\cdot 23 + \left(12 a + 8\right)\cdot 23^{2} + \left(a + 1\right)\cdot 23^{3} + \left(14 a + 18\right)\cdot 23^{4} + \left(9 a + 15\right)\cdot 23^{5} + \left(7 a + 14\right)\cdot 23^{6} + \left(a + 2\right)\cdot 23^{7} + \left(2 a + 15\right)\cdot 23^{8} + \left(13 a + 6\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 11 a + 9 + \left(11 a + 5\right)\cdot 23 + \left(10 a + 14\right)\cdot 23^{2} + \left(21 a + 21\right)\cdot 23^{3} + \left(8 a + 4\right)\cdot 23^{4} + \left(13 a + 7\right)\cdot 23^{5} + \left(15 a + 8\right)\cdot 23^{6} + \left(21 a + 20\right)\cdot 23^{7} + \left(20 a + 7\right)\cdot 23^{8} + \left(9 a + 16\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 12 a + 8 + \left(11 a + 17\right)\cdot 23 + 12 a\cdot 23^{2} + \left(a + 8\right)\cdot 23^{3} + \left(14 a + 1\right)\cdot 23^{4} + \left(9 a + 2\right)\cdot 23^{5} + \left(7 a + 3\right)\cdot 23^{6} + \left(a + 2\right)\cdot 23^{7} + \left(2 a + 5\right)\cdot 23^{8} + \left(13 a + 15\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 17 a + 18 + 7 a\cdot 23 + 15\cdot 23^{2} + \left(2 a + 9\right)\cdot 23^{3} + \left(15 a + 20\right)\cdot 23^{4} + \left(14 a + 15\right)\cdot 23^{5} + \left(12 a + 17\right)\cdot 23^{6} + \left(19 a + 9\right)\cdot 23^{7} + 20 a\cdot 23^{8} + \left(14 a + 7\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 11 a + 16 + \left(11 a + 5\right)\cdot 23 + \left(10 a + 22\right)\cdot 23^{2} + \left(21 a + 14\right)\cdot 23^{3} + \left(8 a + 21\right)\cdot 23^{4} + \left(13 a + 20\right)\cdot 23^{5} + \left(15 a + 19\right)\cdot 23^{6} + \left(21 a + 20\right)\cdot 23^{7} + \left(20 a + 17\right)\cdot 23^{8} + \left(9 a + 7\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.