Properties

Label 2.3600.24t22.c.a
Dimension $2$
Group $\textrm{GL(2,3)}$
Conductor $3600$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $\textrm{GL(2,3)}$
Conductor: \(3600\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Artin stem field: Galois closure of 8.2.139968000000.2
Galois orbit size: $2$
Smallest permutation container: 24T22
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.10800.2

Defining polynomial

$f(x)$$=$ \( x^{8} + 60x^{2} - 300 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: \( x^{2} + 49x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 3 a + 31 + \left(37 a + 18\right)\cdot 53 + \left(14 a + 46\right)\cdot 53^{2} + \left(23 a + 39\right)\cdot 53^{3} + \left(31 a + 49\right)\cdot 53^{4} + \left(33 a + 28\right)\cdot 53^{5} + \left(22 a + 48\right)\cdot 53^{6} + \left(36 a + 47\right)\cdot 53^{7} +O(53^{8})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 50 a + 43 + \left(15 a + 4\right)\cdot 53 + \left(38 a + 15\right)\cdot 53^{2} + \left(29 a + 12\right)\cdot 53^{3} + \left(21 a + 46\right)\cdot 53^{4} + \left(19 a + 25\right)\cdot 53^{5} + \left(30 a + 52\right)\cdot 53^{6} + \left(16 a + 11\right)\cdot 53^{7} +O(53^{8})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 43 + 22\cdot 53 + 29\cdot 53^{2} + 17\cdot 53^{3} + 5\cdot 53^{4} + 37\cdot 53^{5} + 46\cdot 53^{6} + 31\cdot 53^{7} +O(53^{8})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 10 a + 33 + \left(14 a + 29\right)\cdot 53 + \left(14 a + 31\right)\cdot 53^{2} + \left(39 a + 34\right)\cdot 53^{3} + \left(39 a + 19\right)\cdot 53^{4} + \left(26 a + 19\right)\cdot 53^{5} + \left(10 a + 45\right)\cdot 53^{6} + \left(7 a + 43\right)\cdot 53^{7} +O(53^{8})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 50 a + 22 + \left(15 a + 34\right)\cdot 53 + \left(38 a + 6\right)\cdot 53^{2} + \left(29 a + 13\right)\cdot 53^{3} + \left(21 a + 3\right)\cdot 53^{4} + \left(19 a + 24\right)\cdot 53^{5} + \left(30 a + 4\right)\cdot 53^{6} + \left(16 a + 5\right)\cdot 53^{7} +O(53^{8})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 3 a + 10 + \left(37 a + 48\right)\cdot 53 + \left(14 a + 37\right)\cdot 53^{2} + \left(23 a + 40\right)\cdot 53^{3} + \left(31 a + 6\right)\cdot 53^{4} + \left(33 a + 27\right)\cdot 53^{5} + 22 a\cdot 53^{6} + \left(36 a + 41\right)\cdot 53^{7} +O(53^{8})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 10 + 30\cdot 53 + 23\cdot 53^{2} + 35\cdot 53^{3} + 47\cdot 53^{4} + 15\cdot 53^{5} + 6\cdot 53^{6} + 21\cdot 53^{7} +O(53^{8})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 43 a + 20 + \left(38 a + 23\right)\cdot 53 + \left(38 a + 21\right)\cdot 53^{2} + \left(13 a + 18\right)\cdot 53^{3} + \left(13 a + 33\right)\cdot 53^{4} + \left(26 a + 33\right)\cdot 53^{5} + \left(42 a + 7\right)\cdot 53^{6} + \left(45 a + 9\right)\cdot 53^{7} +O(53^{8})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.