Basic invariants
Dimension: | $4$ |
Group: | $C_3^2:D_4$ |
Conductor: | \(3897\)\(\medspace = 3^{2} \cdot 433 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 6.0.11691.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_3^2:D_4$ |
Parity: | even |
Projective image: | $\SOPlus(4,2)$ |
Projective field: | Galois closure of 6.0.11691.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 10 a + 11 + \left(4 a + 7\right)\cdot 13 + \left(4 a + 3\right)\cdot 13^{2} + \left(8 a + 4\right)\cdot 13^{3} + \left(10 a + 11\right)\cdot 13^{4} +O(13^{5})\)
|
$r_{ 2 }$ | $=$ |
\( a + 3 + 9\cdot 13 + \left(10 a + 9\right)\cdot 13^{2} + \left(4 a + 9\right)\cdot 13^{3} + \left(7 a + 2\right)\cdot 13^{4} +O(13^{5})\)
|
$r_{ 3 }$ | $=$ |
\( 3 + 10\cdot 13 + 2\cdot 13^{2} + 2\cdot 13^{3} + 10\cdot 13^{4} +O(13^{5})\)
|
$r_{ 4 }$ | $=$ |
\( 11 + 10\cdot 13^{3} + 8\cdot 13^{4} +O(13^{5})\)
|
$r_{ 5 }$ | $=$ |
\( 12 a + 4 + \left(12 a + 8\right)\cdot 13 + \left(2 a + 6\right)\cdot 13^{2} + \left(8 a + 4\right)\cdot 13^{3} + \left(5 a + 5\right)\cdot 13^{4} +O(13^{5})\)
|
$r_{ 6 }$ | $=$ |
\( 3 a + 8 + \left(8 a + 2\right)\cdot 13 + \left(8 a + 3\right)\cdot 13^{2} + \left(4 a + 8\right)\cdot 13^{3} + 2 a\cdot 13^{4} +O(13^{5})\)
|
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $4$ |
$6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$6$ | $2$ | $(3,5)$ | $2$ |
$9$ | $2$ | $(3,5)(4,6)$ | $0$ |
$4$ | $3$ | $(1,4,6)(2,3,5)$ | $-2$ |
$4$ | $3$ | $(1,4,6)$ | $1$ |
$18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ |
$12$ | $6$ | $(1,3,4,5,6,2)$ | $0$ |
$12$ | $6$ | $(1,4,6)(3,5)$ | $-1$ |