# Properties

 Label 85680fi Number of curves 4 Conductor 85680 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("85680.dp1")

sage: E.isogeny_class()

## Elliptic curves in class 85680fi

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
85680.dp4 85680fi1 [0, 0, 0, 2013, 58466]  131072 $$\Gamma_0(N)$$-optimal
85680.dp3 85680fi2 [0, 0, 0, -15987, 638066] [2, 2] 262144
85680.dp2 85680fi3 [0, 0, 0, -77187, -7672894]  524288
85680.dp1 85680fi4 [0, 0, 0, -242787, 46043426]  524288

## Rank

sage: E.rank()

The elliptic curves in class 85680fi have rank $$1$$.

## Modular form 85680.2.a.dp

sage: E.q_eigenform(10)

$$q + q^{5} - q^{7} - 6q^{13} + q^{17} - 4q^{19} + O(q^{20})$$

## 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 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. 