Properties

Label 4.3215.8t44.b
Dimension $4$
Group $C_2 \wr S_4$
Conductor $3215$
Indicator $1$

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Basic invariants

Dimension:$4$
Group:$C_2 \wr S_4$
Conductor:\(3215\)\(\medspace = 5 \cdot 643 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 8.0.2067245.1
Galois orbit size: $1$
Smallest permutation container: $C_2 \wr S_4$
Parity: odd
Projective image: $C_2^3:S_4$
Projective field: Galois closure of 8.4.258405625.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{2} + 33x + 2 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 23 a + 13 + \left(9 a + 19\right)\cdot 37 + \left(29 a + 6\right)\cdot 37^{2} + \left(28 a + 21\right)\cdot 37^{3} + \left(33 a + 36\right)\cdot 37^{4} + \left(9 a + 20\right)\cdot 37^{5} + \left(20 a + 4\right)\cdot 37^{6} + \left(8 a + 16\right)\cdot 37^{7} + \left(27 a + 8\right)\cdot 37^{8} + \left(35 a + 2\right)\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 6 a + 31 + \left(28 a + 30\right)\cdot 37 + 19 a\cdot 37^{2} + \left(14 a + 21\right)\cdot 37^{3} + \left(7 a + 23\right)\cdot 37^{4} + \left(22 a + 28\right)\cdot 37^{5} + \left(13 a + 23\right)\cdot 37^{6} + \left(17 a + 9\right)\cdot 37^{7} + \left(a + 8\right)\cdot 37^{8} + \left(11 a + 23\right)\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 13 + 11\cdot 37 + 17\cdot 37^{2} + 9\cdot 37^{3} + 8\cdot 37^{4} + 16\cdot 37^{5} + 4\cdot 37^{6} + 26\cdot 37^{7} + 10\cdot 37^{8} + 29\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 14 a + 31 + \left(27 a + 34\right)\cdot 37 + \left(7 a + 2\right)\cdot 37^{2} + \left(8 a + 33\right)\cdot 37^{3} + \left(3 a + 31\right)\cdot 37^{4} + \left(27 a + 26\right)\cdot 37^{5} + \left(16 a + 1\right)\cdot 37^{6} + \left(28 a + 30\right)\cdot 37^{7} + \left(9 a + 34\right)\cdot 37^{8} + \left(a + 6\right)\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 28 a + 11 + \left(32 a + 29\right)\cdot 37 + \left(29 a + 1\right)\cdot 37^{2} + \left(7 a + 25\right)\cdot 37^{3} + \left(28 a + 14\right)\cdot 37^{4} + \left(24 a + 19\right)\cdot 37^{5} + \left(30 a + 34\right)\cdot 37^{6} + 8 a\cdot 37^{7} + \left(33 a + 3\right)\cdot 37^{8} + \left(24 a + 34\right)\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 20 + 11\cdot 37 + 15\cdot 37^{2} + 26\cdot 37^{3} + 22\cdot 37^{4} + 20\cdot 37^{5} + 37^{6} + 31\cdot 37^{7} + 32\cdot 37^{8} + 33\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 31 a + 18 + \left(8 a + 26\right)\cdot 37 + \left(17 a + 14\right)\cdot 37^{2} + \left(22 a + 22\right)\cdot 37^{3} + \left(29 a + 1\right)\cdot 37^{4} + \left(14 a + 36\right)\cdot 37^{5} + \left(23 a + 18\right)\cdot 37^{6} + \left(19 a + 28\right)\cdot 37^{7} + \left(35 a + 33\right)\cdot 37^{8} + \left(25 a + 28\right)\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 9 a + 12 + \left(4 a + 21\right)\cdot 37 + \left(7 a + 14\right)\cdot 37^{2} + \left(29 a + 26\right)\cdot 37^{3} + \left(8 a + 8\right)\cdot 37^{4} + \left(12 a + 16\right)\cdot 37^{5} + \left(6 a + 21\right)\cdot 37^{6} + \left(28 a + 5\right)\cdot 37^{7} + \left(3 a + 16\right)\cdot 37^{8} + \left(12 a + 26\right)\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character values
$c1$
$1$ $1$ $()$ $4$
$1$ $2$ $(1,8)(2,7)(3,6)(4,5)$ $-4$
$4$ $2$ $(2,7)$ $2$
$4$ $2$ $(1,8)(2,7)(3,6)$ $-2$
$6$ $2$ $(2,7)(4,5)$ $0$
$12$ $2$ $(1,3)(2,4)(5,7)(6,8)$ $0$
$12$ $2$ $(1,4)(5,8)$ $2$
$12$ $2$ $(1,8)(2,3)(4,5)(6,7)$ $-2$
$24$ $2$ $(1,4)(2,7)(5,8)$ $0$
$32$ $3$ $(1,3,2)(6,7,8)$ $1$
$12$ $4$ $(1,3,8,6)(2,5,7,4)$ $0$
$12$ $4$ $(1,4,8,5)$ $2$
$12$ $4$ $(1,8)(2,6,7,3)(4,5)$ $-2$
$24$ $4$ $(1,3)(2,5,7,4)(6,8)$ $0$
$24$ $4$ $(1,4,8,5)(2,7)$ $0$
$48$ $4$ $(1,4,3,2)(5,6,7,8)$ $0$
$32$ $6$ $(1,3,2,8,6,7)$ $1$
$32$ $6$ $(1,3,2)(4,5)(6,7,8)$ $-1$
$32$ $6$ $(1,3,2,8,6,7)(4,5)$ $-1$
$48$ $8$ $(1,4,3,2,8,5,6,7)$ $0$
The blue line marks the conjugacy class containing complex conjugation.