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Let O be a quaternion order with base ring Z, Fq[X] with q odd, or a
number ring. Then Magma can test the following predicates.
Returns true if and only if the order O is maximal.
IspMaximal(O, p) : AlgQuatOrd, RngElt -> BoolElt
Returns true if and only if the order O is maximal at the prime or prime ideal p.
MaximalOrders: BoolElt Default: false
Returns true if and only if the order O is Eichler, that is
an intersection of two (not necessarily distinct) maximal orders. The
function calls the EichlerInvariant intrinsic explained below.
If the optional argument MaximalOrders is set to true, the algorithm also
returns two maximal orders such that O is their intersection.
IsEichler(O, p) : AlgQuatOrd , RngElt -> BoolElt, AlgQuatOrd, AlgQuatOrd
MaximalOrders: BoolElt Default: false
Returns true if and only if the completion of the order O at
the prime (ideal) p is Eichler.
If the optional argument MaximalOrders is set to true, the algorithm also
returns two p-maximal orders such that O is their intersection.
EichlerInvariant(O, p) : AlgQuatOrd , RngElt -> RngIntElt
Returns the local Eichler invariant of O at some prime (ideal) p which
divides the discriminant of O. Let R be the base ring of O and let
J be the Jacobson radical of the R/p-algebra O/pO. If J has
dimension 3 then the Eichler invariant is defined to be 0. Otherwise
the quotient of O/pO by J is either isomorphic to a direct sum of
two copies of R/p or a quadratic field extension of R/p. In the first
case the Eichler invariant is 1, in the latter it is -1.
Returns true if and only if the order O is a hereditary order in a
quaternion algebra A. That is, every lattice in A of full rank such
that O is contained in its left order is a projective left O-module.
The hereditary orders are precisely those with squarefree discriminant.
IsHereditary(O, p) : AlgQuatOrd , RngElt -> BoolElt
Returns true if and only if the completion of the order O at
the prime (ideal) p is hereditary.
Returns true if and only if the order O is a Gorenstein order.
That is, the dual of O with respect to the trace bilinear form is a
projective O-module. The second return value is the Brandt invariant
of O as in the GorensteinClosure intrinsic.
IsGorenstein(O, p) : AlgQuatOrd , RngElt -> BoolElt, RngIntElt
Returns true if and only if the completion of the order O at
the prime (ideal) p is Gorenstein. The second return value is
the valuation of the Brandt invariant of O at p.
Returns true if and only if the order O is a Bass order, i.e.
every order which contains O is Gorenstein.
IsBass(O, p) : AlgQuatOrd , RngElt -> BoolElt
Returns true if and only if the completion of the order O at
the prime (ideal) p is Bass.
Returns true if and only if the two quaternion orders O1 and O2
are of the same type which means that they are locally isomorphic.
The orders must be over the ring of integers or a number ring.
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