# Properties

 Label 3249.d Number of curves $3$ Conductor $3249$ CM no Rank $1$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 3249.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3249.d1 3249c3 $$[0, 0, 1, -2499564, -1521053555]$$ $$-50357871050752/19$$ $$-651632497731$$ $$[]$$ $$25920$$ $$2.0550$$
3249.d2 3249c2 $$[0, 0, 1, -30324, -2162300]$$ $$-89915392/6859$$ $$-235239331680891$$ $$[]$$ $$8640$$ $$1.5057$$
3249.d3 3249c1 $$[0, 0, 1, 2166, -1715]$$ $$32768/19$$ $$-651632497731$$ $$[]$$ $$2880$$ $$0.95635$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 3249.d have rank $$1$$.

## Complex multiplication

The elliptic curves in class 3249.d do not have complex multiplication.

## Modular form3249.2.a.d

sage: E.q_eigenform(10)

$$q - 2q^{4} - 3q^{5} - q^{7} - 3q^{11} + 4q^{13} + 4q^{16} + 3q^{17} + O(q^{20})$$ ## 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. 