# Properties

 Label 51870j Number of curves 4 Conductor 51870 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("51870.j1")

sage: E.isogeny_class()

## Elliptic curves in class 51870j

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
51870.j4 51870j1 [1, 1, 0, -69218, -2015628]  589824 $$\Gamma_0(N)$$-optimal
51870.j2 51870j2 [1, 1, 0, -869218, -311935628] [2, 2] 1179648
51870.j3 51870j3 [1, 1, 0, -635218, -483364028]  2359296
51870.j1 51870j4 [1, 1, 0, -13903218, -19959387228]  2359296

## Rank

sage: E.rank()

The elliptic curves in class 51870j have rank $$0$$.

## Modular form 51870.2.a.j

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 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. 