# Properties

 Label 650.f Number of curves $2$ Conductor $650$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 650.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
650.f1 650b2 $$[1, 1, 0, -11330, -468940]$$ $$-6434774386429585/140608$$ $$-3515200$$ $$[]$$ $$1080$$ $$0.78239$$
650.f2 650b1 $$[1, 1, 0, -130, -780]$$ $$-9836106385/3407872$$ $$-85196800$$ $$[]$$ $$360$$ $$0.23308$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 650.f have rank $$1$$.

## Complex multiplication

The elliptic curves in class 650.f do not have complex multiplication.

## Modular form650.2.a.f

sage: E.q_eigenform(10)

$$q - q^{2} + 2 q^{3} + q^{4} - 2 q^{6} - 5 q^{7} - q^{8} + q^{9} - 3 q^{11} + 2 q^{12} - q^{13} + 5 q^{14} + q^{16} - 3 q^{17} - q^{18} - 4 q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 