Basic invariants
Dimension: | $4$ |
Group: | $\PGL(2,5)$ |
Conductor: | \(3233\)\(\medspace = 53 \cdot 61 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 6.2.33792250337.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | even |
Projective image: | $S_5$ |
Projective field: | Galois closure of 6.2.33792250337.1 |
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:
$r_{ 1 }$ | $=$ |
\( 13 + 8\cdot 13^{2} + 10\cdot 13^{3} + 12\cdot 13^{4} +O(13^{5})\)
$r_{ 2 }$ |
$=$ |
\( 5 a + 12 + \left(6 a + 8\right)\cdot 13 + \left(3 a + 6\right)\cdot 13^{2} + \left(12 a + 3\right)\cdot 13^{3} + \left(7 a + 4\right)\cdot 13^{4} +O(13^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 4 + 2\cdot 13 + 8\cdot 13^{2} + 12\cdot 13^{3} + 9\cdot 13^{4} +O(13^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 8 a + 6 + \left(3 a + 10\right)\cdot 13 + \left(7 a + 10\right)\cdot 13^{2} + \left(2 a + 8\right)\cdot 13^{3} + \left(4 a + 11\right)\cdot 13^{4} +O(13^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 8 a + 4 + \left(6 a + 10\right)\cdot 13 + \left(9 a + 3\right)\cdot 13^{2} + 12\cdot 13^{3} + \left(5 a + 12\right)\cdot 13^{4} +O(13^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 5 a + 1 + \left(9 a + 6\right)\cdot 13 + \left(5 a + 1\right)\cdot 13^{2} + \left(10 a + 4\right)\cdot 13^{3} + 8 a\cdot 13^{4} +O(13^{5})\)
| |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $4$ |
$10$ | $2$ | $(1,3)(2,4)(5,6)$ | $2$ |
$15$ | $2$ | $(1,3)(4,6)$ | $0$ |
$20$ | $3$ | $(1,6,5)(2,3,4)$ | $1$ |
$30$ | $4$ | $(1,4,3,6)$ | $0$ |
$24$ | $5$ | $(1,5,4,6,2)$ | $-1$ |
$20$ | $6$ | $(1,4,6,2,5,3)$ | $-1$ |