Basic invariants
Dimension: | $5$ |
Group: | $A_6$ |
Conductor: | \(1971216\)\(\medspace = 2^{4} \cdot 3^{6} \cdot 13^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 6.2.1971216.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_6$ |
Parity: | even |
Projective image: | $A_6$ |
Projective field: | Galois closure of 6.2.1971216.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{2} + 70x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 61 a + 28 + \left(14 a + 25\right)\cdot 73 + \left(57 a + 30\right)\cdot 73^{2} + \left(34 a + 65\right)\cdot 73^{3} + \left(15 a + 71\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 2 }$ | $=$ | \( 60 + 46\cdot 73 + 62\cdot 73^{2} + 17\cdot 73^{3} + 22\cdot 73^{4} +O(73^{5})\) |
$r_{ 3 }$ | $=$ | \( 12 a + 65 + \left(58 a + 8\right)\cdot 73 + \left(15 a + 41\right)\cdot 73^{2} + \left(38 a + 39\right)\cdot 73^{3} + \left(57 a + 10\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 4 }$ | $=$ | \( 61 + 21\cdot 73 + 68\cdot 73^{2} + 67\cdot 73^{3} + 71\cdot 73^{4} +O(73^{5})\) |
$r_{ 5 }$ | $=$ | \( 72 a + 42 + \left(41 a + 67\right)\cdot 73 + \left(27 a + 60\right)\cdot 73^{2} + \left(44 a + 70\right)\cdot 73^{3} + \left(63 a + 20\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 6 }$ | $=$ | \( a + 39 + \left(31 a + 48\right)\cdot 73 + \left(45 a + 28\right)\cdot 73^{2} + \left(28 a + 30\right)\cdot 73^{3} + \left(9 a + 21\right)\cdot 73^{4} +O(73^{5})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $5$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
$40$ | $3$ | $(1,2,3)$ | $2$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$72$ | $5$ | $(1,3,4,5,2)$ | $0$ |