# Properties

 Label 3211.a Number of curves $3$ Conductor $3211$ CM no Rank $1$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 3211.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3211.a1 3211a3 $$[0, 1, 1, -130017, -18088053]$$ $$-50357871050752/19$$ $$-91709371$$ $$[]$$ $$6480$$ $$1.3159$$
3211.a2 3211a2 $$[0, 1, 1, -1577, -26178]$$ $$-89915392/6859$$ $$-33107082931$$ $$[]$$ $$2160$$ $$0.76661$$
3211.a3 3211a1 $$[0, 1, 1, 113, 17]$$ $$32768/19$$ $$-91709371$$ $$[]$$ $$720$$ $$0.21730$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 3211.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 3211.a do not have complex multiplication.

## Modular form3211.2.a.a

sage: E.q_eigenform(10)

$$q - 2q^{3} - 2q^{4} - 3q^{5} + q^{7} + q^{9} - 3q^{11} + 4q^{12} + 6q^{15} + 4q^{16} - 3q^{17} - q^{19} + O(q^{20})$$ ## Isogeny matrix

sage: E.isogeny_class().matrix()

The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the LMFDB numbering.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels. 