# Properties

 Label 51870bd Number of curves 8 Conductor 51870 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("51870.y1")

sage: E.isogeny_class()

## Elliptic curves in class 51870bd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
51870.y7 51870bd1 [1, 0, 1, -322589, 69234536] [6] 663552 $$\Gamma_0(N)$$-optimal
51870.y6 51870bd2 [1, 0, 1, -687089, -116077264] [2, 6] 1327104
51870.y5 51870bd3 [1, 0, 1, -3013004, -1983986998] [2] 1990656
51870.y8 51870bd4 [1, 0, 1, 2278861, -850446484] [6] 2654208
51870.y4 51870bd5 [1, 0, 1, -9485039, -11240205244] [6] 2654208
51870.y2 51870bd6 [1, 0, 1, -48013004, -128055986998] [2, 2] 3981312
51870.y3 51870bd7 [1, 0, 1, -47818004, -129147674998] [2] 7962624
51870.y1 51870bd8 [1, 0, 1, -768208004, -8195392298998] [2] 7962624

## Rank

sage: E.rank()

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

## Modular form 51870.2.a.y

sage: E.q_eigenform(10)

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

## 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 Cremona numbering.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.