Properties

Label 6.1278983549.8t41.a
Dimension $6$
Group $V_4^2:(S_3\times C_2)$
Conductor $1278983549$
Indicator $1$

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Basic invariants

Dimension:$6$
Group:$V_4^2:(S_3\times C_2)$
Conductor:\(1278983549\)\(\medspace = 29^{3} \cdot 229^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 8.4.37090522921.1
Galois orbit size: $1$
Smallest permutation container: $V_4^2:(S_3\times C_2)$
Parity: even
Projective image: $C_2^3:S_4$
Projective field: Galois closure of 8.4.37090522921.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: \( x^{3} + 4x + 64 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 53 a^{2} + 23 a + 44 + \left(59 a^{2} + 11 a + 20\right)\cdot 71 + \left(6 a^{2} + 54 a + 33\right)\cdot 71^{2} + \left(36 a^{2} + 2 a + 14\right)\cdot 71^{3} + \left(32 a^{2} + 66 a + 27\right)\cdot 71^{4} + \left(18 a^{2} + 70 a + 42\right)\cdot 71^{5} + \left(15 a^{2} + 39 a + 25\right)\cdot 71^{6} + \left(34 a^{2} + a + 27\right)\cdot 71^{7} + \left(54 a^{2} + 58 a + 33\right)\cdot 71^{8} + \left(33 a^{2} + 45 a + 49\right)\cdot 71^{9} +O(71^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 35 a^{2} + 41 a + 67 + \left(30 a^{2} + 56 a + 60\right)\cdot 71 + \left(62 a^{2} + 58 a + 15\right)\cdot 71^{2} + \left(52 a^{2} + 36 a + 59\right)\cdot 71^{3} + \left(7 a^{2} + 41 a + 55\right)\cdot 71^{4} + \left(a^{2} + 17 a + 19\right)\cdot 71^{5} + \left(48 a^{2} + 62 a + 18\right)\cdot 71^{6} + \left(13 a^{2} + a + 67\right)\cdot 71^{7} + \left(52 a^{2} + 3 a + 50\right)\cdot 71^{8} + \left(3 a^{2} + 8 a + 16\right)\cdot 71^{9} +O(71^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 40 + 41\cdot 71 + 42\cdot 71^{2} + 37\cdot 71^{3} + 16\cdot 71^{4} + 16\cdot 71^{5} + 32\cdot 71^{6} + 23\cdot 71^{7} + 17\cdot 71^{8} + 28\cdot 71^{9} +O(71^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 13 a^{2} + 58 a + 14 + \left(15 a^{2} + 51 a + 2\right)\cdot 71 + \left(40 a^{2} + 35 a + 61\right)\cdot 71^{2} + \left(59 a^{2} + 69 a + 68\right)\cdot 71^{3} + \left(42 a^{2} + 47 a + 10\right)\cdot 71^{4} + \left(43 a^{2} + 68 a + 59\right)\cdot 71^{5} + \left(2 a^{2} + 37 a + 62\right)\cdot 71^{6} + \left(53 a^{2} + 68 a + 43\right)\cdot 71^{7} + \left(53 a^{2} + 7 a + 59\right)\cdot 71^{8} + \left(23 a^{2} + 21 a + 14\right)\cdot 71^{9} +O(71^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 52 a^{2} + 20 a + 47 + \left(41 a^{2} + 64 a + 25\right)\cdot 71 + \left(63 a^{2} + 23 a + 52\right)\cdot 71^{2} + \left(13 a^{2} + 44 a + 41\right)\cdot 71^{3} + \left(68 a^{2} + 58 a + 7\right)\cdot 71^{4} + \left(32 a^{2} + 18 a + 7\right)\cdot 71^{5} + \left(45 a^{2} + 63 a + 35\right)\cdot 71^{6} + \left(7 a^{2} + 10 a + 17\right)\cdot 71^{7} + \left(47 a^{2} + 14 a + 18\right)\cdot 71^{8} + \left(29 a^{2} + 39 a + 54\right)\cdot 71^{9} +O(71^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 33 + 63\cdot 71 + 50\cdot 71^{2} + 51\cdot 71^{3} + 45\cdot 71^{4} + 33\cdot 71^{5} + 58\cdot 71^{6} + 34\cdot 71^{7} + 71^{8} + 26\cdot 71^{9} +O(71^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 54 a^{2} + 7 a + 23 + \left(51 a^{2} + 3 a + 70\right)\cdot 71 + \left(a^{2} + 29 a + 66\right)\cdot 71^{2} + \left(53 a^{2} + 31 a + 35\right)\cdot 71^{3} + \left(30 a^{2} + 34 a + 22\right)\cdot 71^{4} + \left(51 a^{2} + 53 a + 59\right)\cdot 71^{5} + \left(7 a^{2} + 39 a + 52\right)\cdot 71^{6} + \left(23 a^{2} + 67 a + 68\right)\cdot 71^{7} + \left(35 a^{2} + 9 a + 5\right)\cdot 71^{8} + \left(33 a^{2} + 17 a + 25\right)\cdot 71^{9} +O(71^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 6 a^{2} + 64 a + 19 + \left(14 a^{2} + 25 a + 70\right)\cdot 71 + \left(38 a^{2} + 11 a + 31\right)\cdot 71^{2} + \left(68 a^{2} + 28 a + 45\right)\cdot 71^{3} + \left(30 a^{2} + 35 a + 26\right)\cdot 71^{4} + \left(65 a^{2} + 54 a + 46\right)\cdot 71^{5} + \left(22 a^{2} + 40 a + 69\right)\cdot 71^{6} + \left(10 a^{2} + 62 a\right)\cdot 71^{7} + \left(41 a^{2} + 48 a + 26\right)\cdot 71^{8} + \left(17 a^{2} + 10 a + 69\right)\cdot 71^{9} +O(71^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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