Basic invariants
Dimension: | $5$ |
Group: | $A_6$ |
Conductor: | \(241864704\)\(\medspace = 2^{12} \cdot 3^{10} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 6.2.26873856.3 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_6$ |
Parity: | even |
Projective image: | $A_6$ |
Projective field: | Galois closure of 6.2.26873856.3 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 31 + 53 + 45\cdot 53^{2} + 49\cdot 53^{3} + 17\cdot 53^{4} +O(53^{5})\) |
$r_{ 2 }$ | $=$ | \( 19 a + 40 + \left(14 a + 14\right)\cdot 53 + 40 a\cdot 53^{2} + \left(40 a + 38\right)\cdot 53^{3} + \left(7 a + 16\right)\cdot 53^{4} +O(53^{5})\) |
$r_{ 3 }$ | $=$ | \( 14 a + 23 + \left(49 a + 12\right)\cdot 53 + \left(22 a + 1\right)\cdot 53^{2} + \left(20 a + 9\right)\cdot 53^{3} + \left(32 a + 18\right)\cdot 53^{4} +O(53^{5})\) |
$r_{ 4 }$ | $=$ | \( 29 + 40\cdot 53 + 27\cdot 53^{2} + 45\cdot 53^{3} + 24\cdot 53^{4} +O(53^{5})\) |
$r_{ 5 }$ | $=$ | \( 34 a + 10 + 38 a\cdot 53 + \left(12 a + 41\right)\cdot 53^{2} + \left(12 a + 1\right)\cdot 53^{3} + \left(45 a + 7\right)\cdot 53^{4} +O(53^{5})\) |
$r_{ 6 }$ | $=$ | \( 39 a + 26 + \left(3 a + 36\right)\cdot 53 + \left(30 a + 43\right)\cdot 53^{2} + \left(32 a + 14\right)\cdot 53^{3} + \left(20 a + 21\right)\cdot 53^{4} +O(53^{5})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $5$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
$40$ | $3$ | $(1,2,3)$ | $-1$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$72$ | $5$ | $(1,3,4,5,2)$ | $0$ |