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108 5. Reduction mod p and Torsion Points

2 3 c 4 c 6
y = x − − .
48 864
6
4
By the above remark, if 48p | c 4 and 864p | c 6 , then the equation is not minimal.
4
5
3
Since the equation is minimal and since 48 = 2 · 3 and 864 = 2 · 3 , it follows that
v( ) + v( j) = 3v(c 4 )< 12 + 3v(48) = 12 + 12v(2) + 3v(3),
or
3
v( ) + v( j − 12 ) = 6v(c 6 )< 12 + 2v(864) = 12 + 10v(2) + 6v(3).
3
Since v( ) + min{v( j), 0}≤ v( ) + v( j) or v( ) + v( j − 12 ), we obtain the first
inequality.
For the second statement observe that for v(2) = v(3) = 0, the minimal model
satisfies v( ) + min{v( j), 0} < 12. The converse holds by Remark (2.4) since 0 <
v( ) + min{v( j), 0} and the relation between two valuations of the discriminants
given above (2.3). This proves the proposition.

Now we return to the notations of (1.1) where R is a factorial ring with field
of fractions k. Two normal forms of equations for E with coefficients a j in R are
2
related by an admissible change of variables 3(2.3) where x = u ¯x + r and y =
2
3
u ¯y + su ¯x + t. The discriminants are related by u = by 3(3.2). For an
12 ¯
irreducible element p in R we have
ord p ( ) = 12 ord p (u) + ord p ( ¯ ).
This leads to the global version of (2.2) since by a change of variable we can always
choose an equation where ord p ( ) is minimal for all irreducibles p in R.
(2.6) Definition. Let k be the field of fractions of a factorial ring R, and let E be
an elliptic curve over k. A minimal normal form for E is a normal form with all
a j in R such that all ord p ( ) is minimal among all equations in normal form with
coefficients a j in R.
By the above discussion a minimal normal form for an elliptic curve E always
exists.
Unfortunately we are interested in rings R, namely the ring of integers in a num-
ber field k, which are not always factorial. In the case there is only a minimal model
of an elliptic curve E locally at valuations v of k with valuation ring R (v) associ-
ated with prime ideals of R. For questions of reduction mod v it is good enough to
work locally with coefficients in R (v) . We return later to the more general case of a
Dedekind ring R to define the conductor of an elliptic curve. In any case the present
theory includes curves defined over Q.

Exercise

1. Over the rational numbers Q under what conditions on the constant a are the normal
2
3
3
2
forms y = x + ax and y = x + a minimal.