# Properties

 Label 15680f Number of curves $4$ Conductor $15680$ CM no Rank $2$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("15680.bv1")

sage: E.isogeny_class()

## Elliptic curves in class 15680f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
15680.bv4 15680f1 [0, 0, 0, 7252, 378672]  36864 $$\Gamma_0(N)$$-optimal
15680.bv3 15680f2 [0, 0, 0, -55468, 4066608] [2, 2] 73728
15680.bv2 15680f3 [0, 0, 0, -274988, -51867088]  147456
15680.bv1 15680f4 [0, 0, 0, -839468, 296028208]  147456

## Rank

sage: E.rank()

The elliptic curves in class 15680f have rank $$2$$.

## Modular form 15680.2.a.bv

sage: E.q_eigenform(10)

$$q - q^{5} - 3q^{9} - 4q^{11} - 6q^{13} - 2q^{17} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 