The modular curve X_{0}(37)

Let θ_{A}, θ_{B}, and θ_{C} be the theta-functions associated to the type-II positive-definite quadratic forms with matrices

A = ( \begin{matrix}4 & 0 & 2 & 1 \\ 0 & 2 & 1 & 1 \\ 2 & 1 & 20 & 1 \\ 1 & 1 & 1 & 10 \end{matrix} ) , B = ( \begin{matrix}2 & 1 & 0 & 1 \\ 1 & 8 & 1 & 4 \\ 0 & 1 & 10 & 2 \\ 1 & 4 & 2 & 12 \end{matrix} ) , C = ( \begin{matrix}4 & 1 & 2 & 1 \\ 1 & 4 & 1 & 0 \\ 2 & 1 & 6 & −2 \\ 1 & 0 & −2 & 20 \end{matrix} ) .

These have discriminant 37^{2} and level 37; also A is self-dual and B, C are each other’s duals. Thus we obtain an anti-invariant cusp form ϕ_{−} of weight two:

ϕ_{−} = \frac{1}{4}(3θ_{A} − 2E_{2}) = q + q^{3} − 2q^{4} − q^{7} − 2q^{9} …,

and an invariant cusp-form ϕ_{+} of weight two:

ϕ_{+} = \frac{1}{2}(θ_{B} − θ_{C}) = q − 2q^{2} − 3q^{3} + 2q^{4} − 2q^{5} + 6q^{6} − q^{7} + 6q^{9} …

which is a holomorphic differential on the elliptic curve X_{0}(37)^{+}.

We obtain modular functions on X_{0}(37)^{+} thus: z = (θ_{A}/ϕ_{−}) − 1 is a degree-2 function one of whose poles it at the cusp; since (q dz/dq)/ϕ_{+} = −z^{2} + O(1) at the cusp, we can by symmetry eliminate the other pole of z to construct the degree-2 modular function

x = \frac{1}{2}(z^{2} + 1 − \frac{q dz/dq}{ϕ_{+}} ) = q^{−2} + 2q^{−1} + 5 + 9q + 18q^{2} + 29q^{3} + 51q^{4} + …

on X_{0}(37)^{+}. The C-algebra of modular functions on X_{0}(37)^{+} is then generated by x and

y = (\frac{η}{η_{37}} )^{2} + 37(\frac{η_{37}}{η} )^{2} + 5x − 7 = q^{−3} + 3q^{−2} + 9q^{−1} + 20 + 46q + 92q^{2} + …;

these satisfy the familiar equation

y^{2} + y = x^{3} − x,

and the invariant differential is ϕ_{+}(dq/q) = −dx/(2y + 1).

We then compute that z = (y + 1)/(x + 1) and (ϕ_{−}/ϕ_{+})^{2} = 1 + 4(x − 6)/(y − 6x + 22), to obtain the anti-invariant modular function

s = \frac{x^{2} − 6x − 11 − 4y}{x + 1} · \frac{ϕ_{−}}{ϕ_{+}} = q^{−2} − 2 − 5q − 14q^{2} − 19q^{3} − …,

with

s^{2} = x^{2} − 6x − 11 − 4y.

In terms of these, we find (let ω =ϕ_{−}):

ω^{−1}E_{2} = \frac{3y − x + 2}{2(x + 1)},
\frac{1}{2}ω^{−2}(E_{4} + E′_{4}) = 5 \frac{137x^{2}y − 250x^{3} − 356xy + 189x^{2} + 57y − 23x}{(x + 1)y},
\frac{1}{2}ω^{−2}(E_{4} − E′_{4}) = 12s[\frac{10xy + 100x^{2} − 13y − 18x − 85}{(x + 1)^{2}(x − 1)} − 57],
\frac{1}{2}ω^{−3}(E_{6} + E′_{6}) =

s[\frac{25327x^{3}y − 25579x^{4} − 33643x^{2}y + 31252x^{3} − 19640xy − 1549x^{2} + 18227y − 26005x + 12152}{(x + 1)^{3}(x − 1)}],

\frac{1}{2}ω^{−3}(E_{6} − E′_{6}) = −126 \frac{201x^{5} − 601x^{3}y − 742x^{4} + 414x^{2}y − 703x^{3} + 346xy + 919x^{2} − 3y + 85x}{(x + 1)^{2}y}.

The exotic 37-isogeny over Q

It is well-known that, of the ten integral points on the elliptic curve X_{0}(37)^{+} (those with x-coordinates −1, 1, 0, 2, 6, which are ±3P, ±P, ±2P, ±4P, ±6P respectively where P : (x, y)=(0, 0) is a generator of the infinite cyclic group of rational points on that curve), nine are Heegner points (the quadratic orders of discriminants 3, 4, 7, 11, 12, 16, 27, 28, 67 giving 3P, 2P, −P, P, −2P, −3P, −4P, 4P, 6P respectively) and one, the point −6P : (x, y) = (6, −15), lifts to a pair of rational points on X_{0}(37) that determine the unique pair of rational j-invariants of 37-isogenous elliptic curves. The “remarks on isogenies” preceding the Antwerp tables exhibit a minimal model for a curve of least conductor, namely 1225=35^{2}, for one of these j-invariants; no derivation is given there beyond the attribution to Vélu. We use our explicit modular functions on X_{0}(37) to confirm the equation for that curve, give a minimal model for the isogenous curve, and determine the kernel of the isogeny.

We begin by choosing a quadratic twist; in the notation of the “general formulas” take β = −(35/37). We then obtain, at the two rational points (x, y, s) = (6, −15, 7) and (6, −15, −7) on X_{0}(37), the elliptic curves

E_{1} : Y_{1}^{2} = X_{1}^{3} − \frac{385}{3}X_{1} + \frac{16450}{27}, E_{2} : Y_{1}^{2} = X_{1}^{3} − \frac{9987985}{3}X_{1} − \frac{63131603150}{27};

the transformation (X_{1}, Y_{1}) = (4X + (5/3), 8Y + 4X + 4) then provides the minimal models:

Y^{2} + XY + Y = X^{3} + X^{2} − 8X + 6  (j = −7 · 11^{3})

for E_{1} (this was the equation attributed to Vélu), and

Y^{2} + XY + Y = X^{3} + X^{2} − 208083X − 36621194  (j = −7 · 137^{3}2083^{3})

for E_{2}. Next we compute the kernel K of the 37-isogeny from E_{1} to E_{2}. We find that each of the points of K is defined over the degree-12 real subfield F of the cyclotomic field Q(e^{2πi/35}), and the X-coordinates are defined over its quadratic subfield, which is the totally real sextic number field of least discriminant 300125 = 5^{3}7^{4} obtained by composing the real quadratic and cubic fields of least discriminant: Q(\sqrt{5}) and Q(cos 2π/7). Indeed, K is generated by a 37-torsion point with X-coordinate

X = \sqrt{5}(2 cos \frac{2π}{7} + \frac{1}{2}(3 + \sqrt{5})).

The Galois group of F is canonically isomorphic to the cyclic group (Z/35Z)^{∗}/{±1} of order 12, generated by ±2; the corresponding automorphism of F induces multiplication by 8 on K. This allows the determination mod 37 of the trace of Frobenius of E_{1} mod l for any prime l of good reduction (i.e. l ≠ 5, 7): let α mod 12 be such that p ≡ ±2^{α} mod 35; then the trace mod 37 is 8^{α} + p · 14^{α}. In particular, the trace is divisible by 37 for p = 11, 47, 137, 223, 1543, 1777, 1951, 2971 and no other p < 3000; of these, all are supersingular with the exception of 1951, which has trace −74.