Properties

Label 4.3048516.8t23.b.a
Dimension $4$
Group $\textrm{GL(2,3)}$
Conductor $3048516$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $\textrm{GL(2,3)}$
Conductor: \(3048516\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 97^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.2.31936253616.2
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 4.2.10476.1

Defining polynomial

$f(x)$$=$ \( x^{8} - x^{7} - 5x^{6} + 2x^{5} + 40x^{4} - 106x^{3} + 112x^{2} - 40x - 8 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{2} + 18x + 2 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.