Let
be the maximal
totally real subfield of
,
the cyclotomic field of
-nd
roots of unity. Let
be the
quaternion algebra over
ramified exactly at the unique prime above
and 7 of the real
places of . Let
be a maximal
order in
, and
the Shimura
curve attached to
.
Let
, where
is the unique Atkin–Lehner
involution on
. We
show that the curve
has several striking features. First, it is a hyperelliptic curve of genus
,
whose hyperelliptic involution is exceptional. Second, there are
Weierstrass
points on
,
and exactly half of these points are CM points; they are
defined over the Hilbert class field of the unique CM extension
of class number
contained in
, the cyclotomic
field of
-th
roots of unity. Third, the normal closure of the field of
-torsion of the Jacobian
of
is the Harbater
field
, the unique
Galois number field
unramified outside
and , with Galois
group
. In fact,
the Jacobian
has the remarkable property that each of its simple factors has a
-torsion field whose
normal closure is the field
.
Finally, and perhaps the most striking fact about
, is that it is also
hyperelliptic over
.
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