Let $C$ be an affine curve given by $p_C(x,y)=0$ where $$ p_C=2x^3y + 2xy^3 +x^3 + y^3 + 5x^2y + 5xy^2 + 2x^2 + 2y^2 + 2x^2y^2 + 2xy $$ and let $\overline{C}$ denote the projective closure of $C$. For $K = \mathbb{Q}(\sqrt{-33})$, I am trying to determine all $K$-rational points on $\overline{C}$. Sage/Magma tells us that $\overline{C}$ has genus 1 and is birational (over $\mathbb{Q}$) to the elliptic curve $E_C$ defined by the equation $$ y^2+2xy+2y = x^3 - x^2 - 2x$$ with the map $\phi$ given by $$ \phi : \overline{C} \rightarrow E_C $$ $$ (x,y,z) \mapsto (p_1^{\phi}, p_2^{\phi},p_3^{\phi} ) $$ where $$ p_1^{\phi}=2x^2y^2 + 3x^2yz + 4xy^2z + x^2z^2 + x^2z^2 + 6xyz^2 -y^2z^2 + 2xz^3 $$ $$ p_2^{\phi} = 2x^2y^2 + 4xy^3 - x2^yz + 10xy2^z + 3y^3z - x^2z2^ + 7y2^z2 - 2xz^3 + 2yz^3 $$ $$ p_3^{\phi} = y^4 + 3y^3z + 3y^2z^2 + yz^3$$ By Sage the Mordell-Weil rank is positive and more precisely $E(K) \simeq \mathbb{Z}/6\mathbb{Z} \oplus \mathbb{Z}.$ (In fact, $E_C$ is birational to the modular curve $X_0(36)$ with equation $y^2=x^3+1)$. Does this imply that there are infinitely many $K$-rational points on $\overline{C}$ under $\phi$?
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$\begingroup$ Is the pseudo-inverse of $\phi$ defined over $K$ ? $\endgroup$– NicolastCommented Feb 16, 2021 at 10:16
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1$\begingroup$ Note that any birational morphism between smooth projective curves uniquely extends to an isomorphism, so if $\bar{C}$ is smooth and $\phi$ is well-defined then the rational points on $E_C$ should correspond exactly to those of $\bar{C}$ under $\phi$ $\endgroup$– JefCommented Feb 16, 2021 at 10:35
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1$\begingroup$ I guess $C$ has a singularity at the origin, so it fails to be smooth. But in any case $\bar{C}$ will only differ from $E_C$ at finitely many points so $\bar{C}$ certainly still has infinitely many $K$-rational points $\endgroup$– JefCommented Feb 16, 2021 at 10:38
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$\begingroup$ The curve $\overline{C}$ has exactly two singular points, and each of these points has one place over it defined over $\mathbb{Q}(\sqrt{-3})$. So the $K$-rational points of $\overline{C}$ are in bijection with $E_C(K)$ under $\phi$. $\endgroup$– François BrunaultCommented Feb 16, 2021 at 13:43
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$\begingroup$ Sorry, in my last comment I meant the $K$-rational points of $\overline{C}$ minus the two singular points $(0,0)$ and $(-1,-1)$. $\endgroup$– François BrunaultCommented Feb 16, 2021 at 16:38
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