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> iload "ex3-2.m";
Interactive-loading "ex3-2.m"
> p :=67;
> // define a quaternion algebra
> B<i,j,k> := QuaternionAlgebra< RationalField() | -1,- p >;
> O := MaximalOrder(B);
> Basis(O);
[ 1, i, 1/2*i + 1/2*k, 1/2 + 1/2*j ]
> // discriminant
> Discriminant(O);
67
> assert Discriminant(O) eq Discriminant(B);
> // define a smaller order:
> Ostd := QuaternionOrder([1,i, j,k]);
> // compute the discriminant to see that this order is not maximal
> Discriminant(Ostd), Factorization(Discriminant(Ostd));
268 [ <2, 2>, <67, 1> ]
> // check whether the order is maximal
> IsMaximal(Ostd);
false
> // define another maximal order
> O1 := QuaternionOrder([1, i, (1+k)/2, (i+j)/2]);
> Basis(O1);
[ 1, i, 1/2 + 1/2*k, 1/2*i + 1/2*j ]
> IsMaximal(O1);
true
> IsIsomorphic(O, O1);
true
> // Note that O can be identified with the endomorphism ring of
> // E: y^2 = x^3 - x,
> //O has basis [ 1, i, 1/2*i + 1/2*k, 1/2 + 1/2*j ]
> // whereas O1 can be identified with the endo ring of
> // E1 : y^2 = x^3 + x; which is a quartic twist of E,
> // so the orders need to be isomorphic
> //O1 has basis [ 1, i, 1/2 + 1/2*k, 1/2*i + 1/2*j ]
> Basis(O), Basis(O1);
[ 1, i, 1/2*i + 1/2*k, 1/2 + 1/2*j ]
[ 1, i, 1/2 + 1/2*k, 1/2*i + 1/2*j ]
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