# Properties

 Label 51870k Number of curves 2 Conductor 51870 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("51870.o1")

sage: E.isogeny_class()

## Elliptic curves in class 51870k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
51870.o1 51870k1 [1, 1, 0, -305859437, -2059002924771]  13381632 $$\Gamma_0(N)$$-optimal
51870.o2 51870k2 [1, 1, 0, -302347117, -2108594775779]  26763264

## Rank

sage: E.rank()

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

## Modular form 51870.2.a.o

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{7} - q^{8} + q^{9} - q^{10} - 2q^{11} - q^{12} - q^{13} + q^{14} - q^{15} + q^{16} + 4q^{17} - q^{18} - q^{19} + O(q^{20})$$

## 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. 