# Properties

 Label 51870.b Number of curves 4 Conductor 51870 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("51870.b1")

sage: E.isogeny_class()

## Elliptic curves in class 51870.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
51870.b1 51870e4 [1, 1, 0, -428140743, -3409726086027]  19464192
51870.b2 51870e3 [1, 1, 0, -153375943, 692540686453]  19464192
51870.b3 51870e2 [1, 1, 0, -28614343, -45474082187] [2, 2] 9732096
51870.b4 51870e1 [1, 1, 0, 4153657, -4428885387]  4866048 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Modular form 51870.2.a.b

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 