# Properties

 Label 51842.j Number of curves 6 Conductor 51842 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("51842.j1")

sage: E.isogeny_class()

## Elliptic curves in class 51842.j

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
51842.j1 51842e6 [1, 1, 0, -70777830, -229218601324] [2] 3649536
51842.j2 51842e5 [1, 1, 0, -4420070, -3588945772] [2] 1824768
51842.j3 51842e4 [1, 1, 0, -920735, -279119203] [2] 1216512
51842.j4 51842e2 [1, 1, 0, -272710, 54665514] [2] 405504
51842.j5 51842e1 [1, 1, 0, -13500, 1216412] [2] 202752 $$\Gamma_0(N)$$-optimal
51842.j6 51842e3 [1, 1, 0, 116105, -25508139] [2] 608256

## Rank

sage: E.rank()

The elliptic curves in class 51842.j have rank $$1$$.

## Modular form 51842.2.a.j

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.