Properties

Label 7.116319195136.8t36.a
Dimension $7$
Group $C_2^3:(C_7: C_3)$
Conductor $116319195136$
Indicator $1$

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Basic invariants

Dimension:$7$
Group:$C_2^3:(C_7: C_3)$
Conductor:\(116319195136\)\(\medspace = 2^{12} \cdot 73^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 8.0.116319195136.1
Galois orbit size: $1$
Smallest permutation container: $C_2^3:(C_7: C_3)$
Parity: even
Projective image: $F_8:C_3$
Projective field: Galois closure of 8.0.116319195136.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{3} + 4x + 17 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 12 a^{2} + 11 a + 2 + \left(4 a^{2} + 16 a + 1\right)\cdot 19 + \left(4 a^{2} + 10 a + 11\right)\cdot 19^{2} + \left(18 a^{2} + 11 a + 14\right)\cdot 19^{3} + \left(18 a^{2} + 12 a + 8\right)\cdot 19^{4} + \left(10 a^{2} + 13 a + 5\right)\cdot 19^{5} + \left(a^{2} + 9 a\right)\cdot 19^{6} + \left(a^{2} + 18 a + 14\right)\cdot 19^{7} + \left(14 a^{2} + 8\right)\cdot 19^{8} + \left(17 a^{2} + 6 a + 12\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 2 a^{2} + 17 a + 7 + \left(7 a^{2} + 14 a + 14\right)\cdot 19 + \left(11 a^{2} + 12 a + 2\right)\cdot 19^{2} + \left(4 a^{2} + 13 a + 17\right)\cdot 19^{3} + \left(9 a^{2} + 10 a + 10\right)\cdot 19^{4} + \left(7 a^{2} + 17\right)\cdot 19^{5} + \left(8 a^{2} + 3 a + 18\right)\cdot 19^{6} + \left(9 a^{2} + 10 a + 6\right)\cdot 19^{7} + \left(5 a^{2} + 3 a + 17\right)\cdot 19^{8} + \left(8 a^{2} + 5 a + 6\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 17 a^{2} + 14 a + 9 + \left(15 a^{2} + 6 a + 12\right)\cdot 19 + \left(6 a^{2} + 3 a + 9\right)\cdot 19^{2} + \left(17 a^{2} + 17 a\right)\cdot 19^{3} + \left(11 a^{2} + 15 a + 18\right)\cdot 19^{4} + \left(a^{2} + 6 a + 1\right)\cdot 19^{5} + \left(6 a^{2} + 2 a\right)\cdot 19^{6} + \left(18 a^{2} + 7 a + 18\right)\cdot 19^{7} + \left(2 a^{2} + 12 a + 16\right)\cdot 19^{8} + \left(4 a^{2} + 2 a + 14\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 16 + 10\cdot 19 + 18\cdot 19^{2} + 15\cdot 19^{3} + 13\cdot 19^{4} + 8\cdot 19^{5} + 3\cdot 19^{6} + 6\cdot 19^{7} + 7\cdot 19^{8} + 6\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 15 a + 8 + \left(15 a^{2} + 16 a + 3\right)\cdot 19 + \left(15 a^{2} + 15 a + 10\right)\cdot 19^{2} + \left(6 a^{2} + 10 a + 9\right)\cdot 19^{3} + 6 a^{2} 19^{4} + \left(5 a^{2} + 3 a + 3\right)\cdot 19^{5} + \left(8 a^{2} + 5 a + 18\right)\cdot 19^{6} + \left(3 a^{2} + 16 a + 13\right)\cdot 19^{7} + \left(9 a^{2} + 9 a + 14\right)\cdot 19^{8} + \left(9 a^{2} + 15\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 7 a^{2} + 12 a + 14 + \left(18 a^{2} + 4 a + 18\right)\cdot 19 + \left(17 a^{2} + 11 a + 15\right)\cdot 19^{2} + \left(12 a^{2} + 15 a + 6\right)\cdot 19^{3} + \left(12 a^{2} + 5 a + 17\right)\cdot 19^{4} + \left(2 a^{2} + 2 a + 14\right)\cdot 19^{5} + \left(9 a^{2} + 4 a + 7\right)\cdot 19^{6} + \left(14 a^{2} + 3 a + 5\right)\cdot 19^{7} + \left(14 a^{2} + 8 a + 4\right)\cdot 19^{8} + \left(10 a^{2} + 12 a\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 14 + 17\cdot 19 + 7\cdot 19^{2} + 14\cdot 19^{3} + 13\cdot 19^{5} + 18\cdot 19^{6} + 14\cdot 19^{7} + 13\cdot 19^{8} + 10\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 7 a + 8 + \left(15 a^{2} + 16 a + 16\right)\cdot 19 + \left(2 a + 18\right)\cdot 19^{2} + \left(16 a^{2} + 7 a + 15\right)\cdot 19^{3} + \left(16 a^{2} + 11 a + 5\right)\cdot 19^{4} + \left(9 a^{2} + 11 a + 11\right)\cdot 19^{5} + \left(4 a^{2} + 13 a + 8\right)\cdot 19^{6} + \left(10 a^{2} + a + 15\right)\cdot 19^{7} + \left(10 a^{2} + 3 a + 11\right)\cdot 19^{8} + \left(6 a^{2} + 11 a + 8\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character values
$c1$
$1$ $1$ $()$ $7$
$7$ $2$ $(1,4)(2,6)(3,5)(7,8)$ $-1$
$28$ $3$ $(2,5,8)(3,7,6)$ $1$
$28$ $3$ $(2,8,5)(3,6,7)$ $1$
$28$ $6$ $(1,4)(2,7,5,6,8,3)$ $-1$
$28$ $6$ $(1,4)(2,3,8,6,5,7)$ $-1$
$24$ $7$ $(2,8,6,5,3,7,4)$ $0$
$24$ $7$ $(2,5,4,6,7,8,3)$ $0$
The blue line marks the conjugacy class containing complex conjugation.