# Properties

 Label 51870f Number of curves 2 Conductor 51870 CM no Rank 0 Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("51870.h1")
sage: E.isogeny_class()

## Elliptic curves in class 51870f

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
51870.h2 51870f1 [1, 1, 0, -598, -6188] 2 33280 $$\Gamma_0(N)$$-optimal
51870.h1 51870f2 [1, 1, 0, -9718, -372812] 2 66560

## Rank

sage: E.rank()

The elliptic curves in class 51870f have rank $$0$$.

## Modular form None

sage: E.q_eigenform(10)
$$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + q^{7} - q^{8} + q^{9} + q^{10} - q^{12} - q^{13} - q^{14} + q^{15} + q^{16} + 2q^{17} - q^{18} - q^{19} + O(q^{20})$$

## 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 Cremona 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 Cremona labels. 