The HeuristicEndomorphismDescription does not make it clear the answer is a quaternion algebra in this example:
> C;
Hyperelliptic Curve defined by y^2 = 1/144*x^6 - 1/24*x^4 + 5/72*x^3 - 3/64*x^2
+ 5/384*x - 1/768 over Rational Field
> HeuristicEndomorphismAlgebra(C : Geometric := true);
<[ <2, 1> ], [
<1, 4, [ -1, 1 ], 6, 2>
], <1, 1>, 3>
Associative Algebra of dimension 4 with base ring Rational Field
> HeuristicEndomorphismDescription(C : Geometric := true);
abelian variety of dimension 2 with endomorphism algebra a division algebra of
index 2 and discriminant 6 over QQ
To me, it wasn't clear what the latter output means in isolation, e.g., a quadratic field or a quaternion algebra? Is it a commutative algebra or noncommutative? What does "index 2" for an algebra? I guess index means degree as a CSA, but it also doesn't tell me what the center is. It would be more clear to say something like:
a 4-dimensional central division algebra over QQ of discriminant 6
The HeuristicEndomorphismDescription does not make it clear the answer is a quaternion algebra in this example:
To me, it wasn't clear what the latter output means in isolation, e.g., a quadratic field or a quaternion algebra? Is it a commutative algebra or noncommutative? What does "index 2" for an algebra? I guess index means degree as a CSA, but it also doesn't tell me what the center is. It would be more clear to say something like:
a 4-dimensional central division algebra over QQ of discriminant 6