Properties

Label 4.56205009.8t23.a.a
Dimension $4$
Group $\textrm{GL(2,3)}$
Conductor $56205009$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $\textrm{GL(2,3)}$
Conductor: \(56205009\)\(\medspace = 3^{4} \cdot 7^{4} \cdot 17^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.2.438567685227.1
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 4.2.7803.1

Defining polynomial

$f(x)$$=$ \( x^{8} + 3x^{6} - 9x^{5} + 15x^{4} - 66x^{3} + 147x^{2} - 165x - 3 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 16 + 6\cdot 29 + 2\cdot 29^{2} + 10\cdot 29^{3} + 4\cdot 29^{4} + 28\cdot 29^{5} + 29^{6} +O(29^{7})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 8 a + 25 + 18\cdot 29 + \left(11 a + 26\right)\cdot 29^{2} + \left(6 a + 24\right)\cdot 29^{3} + \left(13 a + 19\right)\cdot 29^{4} + \left(11 a + 5\right)\cdot 29^{5} + \left(11 a + 16\right)\cdot 29^{6} +O(29^{7})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 5 a + 22 + \left(18 a + 10\right)\cdot 29 + \left(28 a + 9\right)\cdot 29^{2} + \left(10 a + 4\right)\cdot 29^{3} + \left(27 a + 13\right)\cdot 29^{4} + \left(16 a + 18\right)\cdot 29^{5} + \left(7 a + 15\right)\cdot 29^{6} +O(29^{7})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 8 a + 10 + \left(11 a + 3\right)\cdot 29 + \left(16 a + 9\right)\cdot 29^{2} + \left(8 a + 7\right)\cdot 29^{3} + \left(9 a + 18\right)\cdot 29^{4} + \left(22 a + 24\right)\cdot 29^{5} + \left(20 a + 16\right)\cdot 29^{6} +O(29^{7})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 21 a + 7 + \left(28 a + 12\right)\cdot 29 + \left(17 a + 23\right)\cdot 29^{2} + \left(22 a + 16\right)\cdot 29^{3} + \left(15 a + 21\right)\cdot 29^{4} + \left(17 a + 20\right)\cdot 29^{5} + \left(17 a + 3\right)\cdot 29^{6} +O(29^{7})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 24 a + 18 + \left(10 a + 9\right)\cdot 29 + 18\cdot 29^{2} + \left(18 a + 1\right)\cdot 29^{3} + \left(a + 23\right)\cdot 29^{4} + \left(12 a + 17\right)\cdot 29^{5} + \left(21 a + 7\right)\cdot 29^{6} +O(29^{7})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 26 + 2\cdot 29 + 5\cdot 29^{2} + 17\cdot 29^{3} + 17\cdot 29^{4} + 18\cdot 29^{5} + 13\cdot 29^{6} +O(29^{7})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 21 a + 21 + \left(17 a + 22\right)\cdot 29 + \left(12 a + 21\right)\cdot 29^{2} + \left(20 a + 4\right)\cdot 29^{3} + \left(19 a + 27\right)\cdot 29^{4} + \left(6 a + 10\right)\cdot 29^{5} + \left(8 a + 11\right)\cdot 29^{6} +O(29^{7})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.