# Properties

 Label 4641.e Number of curves $2$ Conductor $4641$ CM no Rank $1$ Graph # Learn more

Show commands: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 4641.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4641.e1 4641d1 $$[1, 0, 1, -46, 107]$$ $$10431681625/710073$$ $$710073$$ $$$$ $$768$$ $$-0.12872$$ $$\Gamma_0(N)$$-optimal
4641.e2 4641d2 $$[1, 0, 1, 39, 481]$$ $$6804992375/102626433$$ $$-102626433$$ $$$$ $$1536$$ $$0.21785$$

## Rank

sage: E.rank()

The elliptic curves in class 4641.e have rank $$1$$.

## Complex multiplication

The elliptic curves in class 4641.e do not have complex multiplication.

## Modular form4641.2.a.e

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} - q^{4} + q^{6} - q^{7} - 3q^{8} + q^{9} + 4q^{11} - q^{12} + q^{13} - q^{14} - q^{16} - q^{17} + q^{18} - 8q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 