# Families of Elliptic Curves with Prescribed $n$-Torsion

In [[Rubin and Silverberg, 1995](../../references.md#cite-rubin-silverberg)], Rubin and Silverberg explicitly construct families of elliptic curves over ${\mathbb{Q}}$ which have the same Galois representation on the $n$-torsion subgroups as a given elliptic curve.

## `RubinSilverbergPolynomials(n, J : parameters): RngIntElt, RngElt -> RngElt, RngElt`

```magma
Parameter: RngElt                    Default: 
```

Suppose that $n = 2$, 3, 4, or 5 and let $E : y^2 = x^3 + ax + b$ be an elliptic curve over the rationals with $j$-invariant $1728 J$. This function returns polynomials $\alpha(t)$ and $\beta(t)$ that determine a family of elliptic curves with fixed $n$-torsion, in the following sense: Every nonsingular member $F_t$ of the family $F : y^2  = x^3 + a\alpha(t) x + b\beta(t)$ has $F_t[n]$ isomorphic to $E[n]$ as ${\mathbb{Z}}[G]$-modules, where $G$ is the absolute Galois group of ${\mathbb{Q}}$, and furthermore the isomorphisms between $F_t[n]$ and $E[n]$ preserve the Weil pairing. When $n$ is 3, 4, or 5, all such “$n$-congruent” curves belong to the same family.