# Properties

 Label 51870a Number of curves 4 Conductor 51870 CM no Rank 2 Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("51870.d1")
sage: E.isogeny_class()

## Elliptic curves in class 51870a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
51870.d4 51870a1 [1, 1, 0, 872, 17728] 2 61440 $$\Gamma_0(N)$$-optimal
51870.d3 51870a2 [1, 1, 0, -7128, 188928] 4 122880
51870.d2 51870a3 [1, 1, 0, -33728, -2221032] 2 245760
51870.d1 51870a4 [1, 1, 0, -108528, 13715688] 2 245760

## Rank

sage: E.rank()

The elliptic curves in class 51870a have rank $$2$$.

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 