# Properties

 Label 51870.p Number of curves 2 Conductor 51870 CM no Rank 1 Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("51870.p1")

sage: E.isogeny_class()

## Elliptic curves in class 51870.p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
51870.p1 51870p1 [1, 1, 0, -48167, 4048821] [2] 256000 $$\Gamma_0(N)$$-optimal
51870.p2 51870p2 [1, 1, 0, -47047, 4247509] [2] 512000

## Rank

sage: E.rank()

The elliptic curves in class 51870.p have rank $$1$$.

## Modular form 51870.2.a.p

sage: E.q_eigenform(10)

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

## 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 LMFDB 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 LMFDB labels.