Properties

Label 4.9801.8t23.b.a
Dimension $4$
Group $\textrm{GL(2,3)}$
Conductor $9801$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $\textrm{GL(2,3)}$
Conductor: \(9801\)\(\medspace = 3^{4} \cdot 11^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.2.32019867.2
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 4.2.3267.1

Defining polynomial

$f(x)$$=$ \( x^{8} - x^{7} + x^{6} - 2x^{5} - x^{4} - 2x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.

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

Roots:
$r_{ 1 }$ $=$ \( 10 a + 27 + \left(19 a + 14\right)\cdot 29 + \left(23 a + 15\right)\cdot 29^{2} + \left(16 a + 12\right)\cdot 29^{3} + \left(8 a + 14\right)\cdot 29^{4} + \left(23 a + 8\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 4 a + 5 + \left(13 a + 14\right)\cdot 29 + \left(26 a + 25\right)\cdot 29^{2} + \left(4 a + 11\right)\cdot 29^{3} + \left(6 a + 14\right)\cdot 29^{4} + \left(7 a + 28\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 24 a + 17 + \left(5 a + 12\right)\cdot 29 + \left(23 a + 14\right)\cdot 29^{2} + 17\cdot 29^{3} + \left(7 a + 27\right)\cdot 29^{4} + \left(5 a + 27\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 7 + 2\cdot 29 + 25\cdot 29^{2} + 5\cdot 29^{3} + 4\cdot 29^{4} + 17\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 25 + 21\cdot 29 + 28\cdot 29^{2} + 15\cdot 29^{3} + 28\cdot 29^{4} + 14\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 5 a + 21 + \left(23 a + 17\right)\cdot 29 + \left(5 a + 8\right)\cdot 29^{2} + \left(28 a + 27\right)\cdot 29^{3} + \left(21 a + 3\right)\cdot 29^{4} + \left(23 a + 18\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 19 a + 19 + \left(9 a + 14\right)\cdot 29 + \left(5 a + 27\right)\cdot 29^{2} + \left(12 a + 14\right)\cdot 29^{3} + \left(20 a + 11\right)\cdot 29^{4} + 5 a\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 25 a + 25 + \left(15 a + 17\right)\cdot 29 + \left(2 a + 28\right)\cdot 29^{2} + \left(24 a + 9\right)\cdot 29^{3} + \left(22 a + 11\right)\cdot 29^{4} + 21 a\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.