Basic invariants
Dimension: | $4$ |
Group: | $C_3^2:C_4$ |
Conductor: | \(1458000\)\(\medspace = 2^{4} \cdot 3^{6} \cdot 5^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 6.2.7290000.2 |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_3^2:C_4$ |
Parity: | even |
Projective image: | $C_3^2:C_4$ |
Projective field: | Galois closure of 6.2.7290000.2 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 15 a + 11 + \left(5 a + 27\right)\cdot 31 + \left(13 a + 12\right)\cdot 31^{2} + 30 a\cdot 31^{3} + \left(17 a + 21\right)\cdot 31^{4} + \left(19 a + 1\right)\cdot 31^{5} + \left(21 a + 20\right)\cdot 31^{6} + \left(23 a + 3\right)\cdot 31^{7} + \left(12 a + 18\right)\cdot 31^{8} + \left(26 a + 16\right)\cdot 31^{9} +O(31^{10})\)
$r_{ 2 }$ |
$=$ |
\( 30 a + 18 + \left(28 a + 8\right)\cdot 31 + \left(30 a + 1\right)\cdot 31^{2} + \left(19 a + 27\right)\cdot 31^{3} + \left(9 a + 24\right)\cdot 31^{4} + \left(17 a + 9\right)\cdot 31^{5} + \left(15 a + 30\right)\cdot 31^{6} + \left(24 a + 8\right)\cdot 31^{7} + \left(23 a + 21\right)\cdot 31^{8} + \left(21 a + 8\right)\cdot 31^{9} +O(31^{10})\)
| $r_{ 3 }$ |
$=$ |
\( a + 16 + \left(2 a + 5\right)\cdot 31 + 3\cdot 31^{2} + \left(11 a + 5\right)\cdot 31^{3} + \left(21 a + 24\right)\cdot 31^{4} + \left(13 a + 3\right)\cdot 31^{5} + \left(15 a + 13\right)\cdot 31^{6} + \left(6 a + 11\right)\cdot 31^{7} + \left(7 a + 13\right)\cdot 31^{8} + \left(9 a + 28\right)\cdot 31^{9} +O(31^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 16 a + 10 + \left(25 a + 23\right)\cdot 31 + \left(17 a + 2\right)\cdot 31^{2} + 17\cdot 31^{3} + \left(13 a + 26\right)\cdot 31^{4} + \left(11 a + 22\right)\cdot 31^{5} + \left(9 a + 12\right)\cdot 31^{6} + \left(7 a + 29\right)\cdot 31^{7} + \left(18 a + 19\right)\cdot 31^{8} + \left(4 a + 25\right)\cdot 31^{9} +O(31^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 10 + 11\cdot 31 + 15\cdot 31^{2} + 13\cdot 31^{3} + 14\cdot 31^{4} + 6\cdot 31^{5} + 29\cdot 31^{6} + 28\cdot 31^{7} + 23\cdot 31^{8} + 19\cdot 31^{9} +O(31^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 28 + 16\cdot 31 + 26\cdot 31^{2} + 29\cdot 31^{3} + 12\cdot 31^{4} + 17\cdot 31^{5} + 18\cdot 31^{6} + 10\cdot 31^{7} + 27\cdot 31^{8} + 24\cdot 31^{9} +O(31^{10})\)
| |
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character values |
$c1$ | |||
$1$ | $1$ | $()$ | $4$ |
$9$ | $2$ | $(1,4)(2,3)$ | $0$ |
$4$ | $3$ | $(2,3,6)$ | $-2$ |
$4$ | $3$ | $(1,4,5)(2,3,6)$ | $1$ |
$9$ | $4$ | $(1,2,4,3)(5,6)$ | $0$ |
$9$ | $4$ | $(1,3,4,2)(5,6)$ | $0$ |