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I am looking for ideals $I\subset \mathbb{F}_2[x,y]$ with the following properties:

1. $I$ is generated by two homogeneous elements;
2. $I$ is invariant under the $SL_2(\mathbb{F}_2)$-action on $\mathbb{F}_2[x,y]$ (given by extending the action on the two dimensional sub vector space spanned by $x,y$).;
3. The quotient $\mathbb{F}_2[x,y]/I$ is finite.

So far the only examples I know are $(x^n,y^n)$ and $(x^3,x^2+xy+y^2)$. The quotient $\mathbb{F}_2[x,y]/(x^3,x^2+xy+y^2)$ is the cohomology ring $H^*(S^3/Q_8;\mathbb{F}_2)$ for the standard action of the quaternion group on the three sphere.

Are these the only examples. Is it possible to classify all such ideals ?

Edit: Of course $(x^n,y^n)$ is only invariant for $n=2$ (since $x\mapsto x, y\mapsto x+y$ is also an automorphism). So we have even fewer examples.

I am looking for ideals $I\subset \mathbb{F}_2[x,y]$ with the following properties:

1. $I$ is generated by two homogeneous elements;
2. $I$ is invariant under the $SL_2(\mathbb{F}_2)$-action on $\mathbb{F}_2[x,y]$ (given by extending the action on the two dimensional sub vector space spanned by $x,y$).;
3. The quotient $\mathbb{F}_2[x,y]/I$ is finite.

So far the only examples I know are $(x^n,y^n)$ and $(x^3,x^2+xy+y^2)$. The quotient $\mathbb{F}_2[x,y]/(x^3,x^2+xy+y^2)$ is the cohomology ring $H^*(S^3/Q_8;\mathbb{F}_2)$ for the standard action of the quaternion group on the three sphere.

Are these the only examples. Is it possible to classify all such ideals ?

Edit: Of course $(x^n,y^n)$ is only invariant when $n$ is a power of two (since $x\mapsto x, y\mapsto x+y$ is also an automorphism). So we have even fewer examples.

I am looking for ideals $I\subset \mathbb{F}_2[x,y]$ with the following properties:

1. $I$ is generated by two homogeneous elements;
2. $I$ is invariant under the $SL_2(\mathbb{F}_2)$-action on $\mathbb{F}_2[x,y]$ (given by extending the action on the two dimensional sub vector space spanned by $x,y$).;
3. The quotient $\mathbb{F}_2[x,y]/I$ is finite.

So far the only examples I know are $(x^n,y^n)$ and $(x^3,x^2+xy+y^2)$. The quotient $\mathbb{F}_2[x,y]/(x^3,x^2+xy+y^2)$ is the cohomology ring $H^*(S^3/Q_8;\mathbb{F}_2)$ for the standard action of the quaternion group on the three sphere.

Are these the only examples. Is it possible to classify all such ideals ?

I am looking for ideals $I\subset \mathbb{F}_2[x,y]$ with the following properties:

1. $I$ is generated by two homogeneous elements;
2. $I$ is invariant under the $SL_2(\mathbb{F}_2)$-action on $\mathbb{F}_2[x,y]$ (given by extending the action on the two dimensional sub vector space spanned by $x,y$).;
3. The quotient $\mathbb{F}_2[x,y]/I$ is finite.

So far the only examples I know are $(x^n,y^n)$ and $(x^3,x^2+xy+y^2)$. The quotient $\mathbb{F}_2[x,y]/(x^3,x^2+xy+y^2)$ is the cohomology ring $H^*(S^3/Q_8;\mathbb{F}_2)$ for the standard action of the quaternion group on the three sphere.

Are these the only examples. Is it possible to classify all such ideals ?

Edit: Of course $(x^n,y^n)$ is only invariant for $n=2$ (since $x\mapsto x, y\mapsto x+y$ is also an automorphism). So we have even fewer examples.

I am looking for ideals $I\subset \mathbb{F}_2[x,y]$ with the following properties:

1. $I$ is generated by two homogeneous elements;
2. $I$ is invariant under the $SL_2(\mathbb{F}_2)$-action on $\mathbb{F}_2$ (given by extending the action on the two dimensional sub vector space spanned by $x,y$).;
3. The quotient $\mathbb{F}_2[x,y]/I$ is finite.

So far the only examples I know are $(x^n,y^n)$ and $(x^3,x^2+xy+y^2)$. The quotient $\mathbb{F}_2[x,y]/(x^3,x^2+xy+y^2)$ is the cohomology ring $H^*(S^3/Q_8;\mathbb{F}_2)$ for the standard action of the quaternion group on the three sphere.

Are these the only examples. Is it possible to classify all such ideals ?

I am looking for ideals $I\subset \mathbb{F}_2[x,y]$ with the following properties:

1. $I$ is generated by two homogeneous elements;
2. $I$ is invariant under the $SL_2(\mathbb{F}_2)$-action on $\mathbb{F}_2[x,y]$ (given by extending the action on the two dimensional sub vector space spanned by $x,y$).;
3. The quotient $\mathbb{F}_2[x,y]/I$ is finite.

So far the only examples I know are $(x^n,y^n)$ and $(x^3,x^2+xy+y^2)$. The quotient $\mathbb{F}_2[x,y]/(x^3,x^2+xy+y^2)$ is the cohomology ring $H^*(S^3/Q_8;\mathbb{F}_2)$ for the standard action of the quaternion group on the three sphere.

Are these the only examples. Is it possible to classify all such ideals ?