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(1.4) Determination of the parity of the modular degreeLet p, E 0 , J, and n be as in Section 1, and fix notation as in Section 1.1. In this section, we prove the following theorem. at University of Manchester on April 13, 2015 http://imrn.oxfordjournals.org/ Downloaded from Modular Parametrizations of Neumann-Setzer Elliptic Curves 1397Proposition 3.2. The curve E 1 is X 1 (p)-optimal.Proof. By [16, Section 5, Lemma 3], E 0 is an optimal quotient of X 0 (p), so we have an). As in [14, page 100], let Σ be the kernel of the functorial map at University of Manchester on April 13, 2015 http://imrn.oxfordjournals.org/ Downloaded from Modular Parametrizations of Neumann-Setzer Elliptic Curves 1401J 0 (p) → J 1 (p) induced by the cover X 1 (p) → X 0 (p). By [14, Proposition II.11.6], Σ is the Cartier dual of the constant subgroup scheme U, which turns out to equal J 0 (p)(Q) tor .Because #(E 0 ∩ U) = 2 and E 0 [2] is self-dual, we have #(E 0 ∩ Σ) = 2. Thus the image of E 0 in J 1 (p) is the quotient of E 0 by the subgroup generated by the rational point of order 2 (note that the Cartier dual of Z/2Z is µ 2 = Z/2Z). This quotient is E 1 , so E 1 ⊂ J 1 (p), which implies that E 1 is an optimal quotient of X 1 (p), as claimed.Remark 3.3. The above proposition could also be proved in a slightly different manner.The Faltings height of an elliptic curve is 2π/Ω, where Ω is the volume of the fundamental parallelogram associated to the curve. When the conductor is prime, we have by [1] that the Manin constants for X 0 (p) and X 1 (p) are 1; this says that for a G-optimal curve E, the period lattice generated by G has covolume equal to Ω E . Since the lattice generated by Γ 1 (p) is contained in the lattice generated by Γ 0 (p) (and thus has larger covolume), the Faltings height of the X 1 (p)-optimal curve must be less than or equal to that of the X 0 (p)optimal curve. So if these two curves differ, the X 1 (p)-optimal curve must have smaller Faltings height.Remark 3.4. In [13, page 12], there is a "to be removed from the final draft" comment that asks (in our notation) whether E 0 is X 0 (p)-optimal when p ≡ 1(mod 16). This is already answered by [16], whereas here we go further and show additionally that E 1 is X 1 (p)optimal. Conjectures Refinement of Theorem 2.1The following conjectural refinement of Theorem 2.1 is supported by the experimental data of [27]. It is unclear whether the method of the proof of Theorem 2.1 can be extended to prove this conjecture. Conjecture 4.1. If u ≡ 7(mod 8), then 2 exactly divides the modular degree of E 0 if and only if u ≡ 7(mod 16). We can note that the pattern seems to end here; for curves with u ≡ 15(mod 16), the data give no further information about the 2-valuation of the modular degree. For instance, with u = −17, we have that the curve defined by [1, 1, 1, −2, 16] has modular degree 2 3 · 3, while with u = 175, the curve [1, 1, 1, −634, −6484] has modular degree 2 2 · 3 3 · 5 · 23. Similarly, we have that u = −33 gives the curve [1, −1, 1, −19, 68] with modular degree 2 5 ·3, whil...

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“…Mestre and Oesterlè also showed that if E tors is ‫,ޚ2/ޚ‬ then E is a Neumann-Setzer curve and N = u 2 + 64. Stein and Watkins [2004] have studied the parity of congruence numbers of Neumann-Setzer curves and they show that E has odd congruence number if and only if u ≡ 3 (mod 8). Furthermore one can show that Neumann-Setzer curves have rank 0 using descent.…”