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 $\mathbb{Q}_{ 19 }$:
\( x^{3} + 4x + 17 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 12 a^{2} + 11 a + 2 + \left(4 a^{2} + 16 a + 1\right)\cdot 19 + \left(4 a^{2} + 10 a + 11\right)\cdot 19^{2} + \left(18 a^{2} + 11 a + 14\right)\cdot 19^{3} + \left(18 a^{2} + 12 a + 8\right)\cdot 19^{4} + \left(10 a^{2} + 13 a + 5\right)\cdot 19^{5} + \left(a^{2} + 9 a\right)\cdot 19^{6} + \left(a^{2} + 18 a + 14\right)\cdot 19^{7} + \left(14 a^{2} + 8\right)\cdot 19^{8} + \left(17 a^{2} + 6 a + 12\right)\cdot 19^{9} +O(19^{10})\)
$r_{ 2 }$ |
$=$ |
\( 2 a^{2} + 17 a + 7 + \left(7 a^{2} + 14 a + 14\right)\cdot 19 + \left(11 a^{2} + 12 a + 2\right)\cdot 19^{2} + \left(4 a^{2} + 13 a + 17\right)\cdot 19^{3} + \left(9 a^{2} + 10 a + 10\right)\cdot 19^{4} + \left(7 a^{2} + 17\right)\cdot 19^{5} + \left(8 a^{2} + 3 a + 18\right)\cdot 19^{6} + \left(9 a^{2} + 10 a + 6\right)\cdot 19^{7} + \left(5 a^{2} + 3 a + 17\right)\cdot 19^{8} + \left(8 a^{2} + 5 a + 6\right)\cdot 19^{9} +O(19^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 17 a^{2} + 14 a + 9 + \left(15 a^{2} + 6 a + 12\right)\cdot 19 + \left(6 a^{2} + 3 a + 9\right)\cdot 19^{2} + \left(17 a^{2} + 17 a\right)\cdot 19^{3} + \left(11 a^{2} + 15 a + 18\right)\cdot 19^{4} + \left(a^{2} + 6 a + 1\right)\cdot 19^{5} + \left(6 a^{2} + 2 a\right)\cdot 19^{6} + \left(18 a^{2} + 7 a + 18\right)\cdot 19^{7} + \left(2 a^{2} + 12 a + 16\right)\cdot 19^{8} + \left(4 a^{2} + 2 a + 14\right)\cdot 19^{9} +O(19^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 16 + 10\cdot 19 + 18\cdot 19^{2} + 15\cdot 19^{3} + 13\cdot 19^{4} + 8\cdot 19^{5} + 3\cdot 19^{6} + 6\cdot 19^{7} + 7\cdot 19^{8} + 6\cdot 19^{9} +O(19^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 15 a + 8 + \left(15 a^{2} + 16 a + 3\right)\cdot 19 + \left(15 a^{2} + 15 a + 10\right)\cdot 19^{2} + \left(6 a^{2} + 10 a + 9\right)\cdot 19^{3} + 6 a^{2} 19^{4} + \left(5 a^{2} + 3 a + 3\right)\cdot 19^{5} + \left(8 a^{2} + 5 a + 18\right)\cdot 19^{6} + \left(3 a^{2} + 16 a + 13\right)\cdot 19^{7} + \left(9 a^{2} + 9 a + 14\right)\cdot 19^{8} + \left(9 a^{2} + 15\right)\cdot 19^{9} +O(19^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 7 a^{2} + 12 a + 14 + \left(18 a^{2} + 4 a + 18\right)\cdot 19 + \left(17 a^{2} + 11 a + 15\right)\cdot 19^{2} + \left(12 a^{2} + 15 a + 6\right)\cdot 19^{3} + \left(12 a^{2} + 5 a + 17\right)\cdot 19^{4} + \left(2 a^{2} + 2 a + 14\right)\cdot 19^{5} + \left(9 a^{2} + 4 a + 7\right)\cdot 19^{6} + \left(14 a^{2} + 3 a + 5\right)\cdot 19^{7} + \left(14 a^{2} + 8 a + 4\right)\cdot 19^{8} + \left(10 a^{2} + 12 a\right)\cdot 19^{9} +O(19^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 14 + 17\cdot 19 + 7\cdot 19^{2} + 14\cdot 19^{3} + 13\cdot 19^{5} + 18\cdot 19^{6} + 14\cdot 19^{7} + 13\cdot 19^{8} + 10\cdot 19^{9} +O(19^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 7 a + 8 + \left(15 a^{2} + 16 a + 16\right)\cdot 19 + \left(2 a + 18\right)\cdot 19^{2} + \left(16 a^{2} + 7 a + 15\right)\cdot 19^{3} + \left(16 a^{2} + 11 a + 5\right)\cdot 19^{4} + \left(9 a^{2} + 11 a + 11\right)\cdot 19^{5} + \left(4 a^{2} + 13 a + 8\right)\cdot 19^{6} + \left(10 a^{2} + a + 15\right)\cdot 19^{7} + \left(10 a^{2} + 3 a + 11\right)\cdot 19^{8} + \left(6 a^{2} + 11 a + 8\right)\cdot 19^{9} +O(19^{10})\)
| |
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character values |
$c1$ | |||
$1$ | $1$ | $()$ | $7$ |
$7$ | $2$ | $(1,4)(2,6)(3,5)(7,8)$ | $-1$ |
$28$ | $3$ | $(2,5,8)(3,7,6)$ | $1$ |
$28$ | $3$ | $(2,8,5)(3,6,7)$ | $1$ |
$28$ | $6$ | $(1,4)(2,7,5,6,8,3)$ | $-1$ |
$28$ | $6$ | $(1,4)(2,3,8,6,5,7)$ | $-1$ |
$24$ | $7$ | $(2,8,6,5,3,7,4)$ | $0$ |
$24$ | $7$ | $(2,5,4,6,7,8,3)$ | $0$ |