# Properties

 Label 720.f Number of curves $4$ Conductor $720$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("f1")

sage: E.isogeny_class()

## Elliptic curves in class 720.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
720.f1 720e3 [0, 0, 0, -1947, -33046]  512
720.f2 720e4 [0, 0, 0, -1227, 16346]  512
720.f3 720e2 [0, 0, 0, -147, -286] [2, 2] 256
720.f4 720e1 [0, 0, 0, 33, -34]  128 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 720.f have rank $$1$$.

## Complex multiplication

The elliptic curves in class 720.f do not have complex multiplication.

## Modular form720.2.a.f

sage: E.q_eigenform(10)

$$q + q^{5} - 4q^{7} - 6q^{13} + 2q^{17} - 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 