# The Smallest Conductor for an Elliptic Curve of Rank Four is Composite

#### August 20, 2003

### Computations

From the work of Stein-Watkins,
an initial database was made of elliptic curves with discriminant *D* of
absolute value less than or equal to 10^{12}, with conductor either
less than 10^{8} or prime conductor less than 10^{10}. However,
this list was not verified to be complete, due to concerns such as minimal quadratic
twists. The approach was to fix minimal *c*-invariants of the elliptic
curve, modulo certain conditions, such that 1728*D*=*c*^{3}_{4}-*c*^{2}_{6}.
*c*_{4} was chosen to be not more than 1.44 × 10^{12}.
However, this bound on the *c*_{4}, as via the ABC Conjecture,
did not necessarily guarantee that the list would be comprehensive, which presented
us with our task.

Below is a graph of the *c*_{4} invariants of all
elliptic curves of conductor *p* found by Stein-Watkins:

Our project stemmed from an implementation of Mestre's
algorithm in MAGMA on
the supercomputers Meccah
and Neron. The final result
of this investigation, then, whether new elliptic curves were to be found or
not, would be a "proof by computer." The material presented henceforth are the
results of these computations, and as such, assume that the computations ran
without any errors due to software or hardware bugs.