# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("51870.q1")

sage: E.isogeny_class()

## Elliptic curves in class 51870l

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
51870.q3 51870l1 [1, 1, 0, -644672, -199409616]  884736 $$\Gamma_0(N)$$-optimal
51870.q2 51870l2 [1, 1, 0, -757172, -125182116] [2, 2] 1769472
51870.q4 51870l3 [1, 1, 0, 2587078, -927133266]  3538944
51870.q1 51870l4 [1, 1, 0, -5901422, 5429579034]  3538944

## Rank

sage: E.rank()

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

## Modular form 51870.2.a.q

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. 