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I would like the equation of a non-isotrivial elliptic curve over the rational function field $\mathbb{F}_2(t)$ with exactly one place of supersingular reduction and I would like to know which place.

I tried $$ Y^2 + tY = X^3 + tX + (t+1) \, . $$ I think it is supersingular at the place $t+1$. Because $t \equiv 1 \bmod {t+1}$ and so the curve reduces at $t+1$ to a curve of equation $Y^2+Y=X^3+X$ which is known to be supersingular. However this curve is isotrivial: its $j$-invariant is zero. I would like a non-isotrivial elliptic curve.

Viktor Vaughn
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user12770
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    What have you tried? – DanLewis3264 Apr 04 '22 at 15:14
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    $Y^2+tY=X^3+tX+(t+1)$…I think it is supersingular at the place t+1. Because $t\cong 1$ mod (t+1) and so the curve reduce at t+1 to a curve of equation $Y^2+Y=X^3+X$ which is known to be supersingular. Is that ok? – user12770 Apr 04 '22 at 16:06
  • Isn't $t \equiv -1 \pmod{t + 1}$? But that strategy seems like it should work – DanLewis3264 Apr 04 '22 at 16:12
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    In characteristic 2, -1=1 – user12770 Apr 04 '22 at 16:15
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    However this curve is isotrivial (its j-invariant is zero). I would like a non-isotrivial elliptic curve. – user12770 Apr 04 '22 at 17:51
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    How about $y^2 + t xy + y = x^3$? Its $j$-invariant is $t^{12}/(t^3 + 1)$. Since the only supersingular $j$-invariant over $\mathbb{F}_2$ is $0$, so I think that shows the only place of supersingular reduction is $t=0$. – Viktor Vaughn Apr 04 '22 at 21:44
  • Umm…If you mod your equation the ideal (t), it gives an equation $y^2+y=x^3$ but this is singular according to [Silverman], AppendixA, prop1.1 second case of part (c) ($\Delta=a_4=0)$. So your curve does not seem to have good supersingular reduction at $t$…Or am I missing something? – user12770 Apr 05 '22 at 09:19
  • @user12770 If you think it is singular, then where is the singular point? The partial derivative with respect to $y$ is $1$, which never vanishes. And the formula in Appendix A of Silverman has a typo; see p. 31 of the errata: https://www.math.brown.edu/johsilve/AEC/AECErrata.pdf But there is no need to look up a formula in a book: one can simply compute the partial derivatives. – Viktor Vaughn Apr 05 '22 at 09:48
  • Thank you very much! It answers totally my question! – user12770 Apr 05 '22 at 11:17
  • @user12770 I posted an answer below. If it answers your question, consider accepting it with the green checkmark. – Viktor Vaughn Apr 12 '22 at 23:57

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Consider the elliptic curve $E: y^2 + t xy + y = x^3 $. Its $j$-invariant is $\frac{t^{12}}{t^3+1}$, so it is not isotrivial. Since the only supersingular elliptic curve over $\overline{\mathbb{F}_2}$ is $y^2 + y = x^3$ with $j$-invariant $0$ (cf., Exercise 5.7 in Silverman's Arithmetic of Elliptic Curves), this shows that the only place of supersingular reduction is $t=0$.

Viktor Vaughn
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