# Properties

 Label 51870m Number of curves 4 Conductor 51870 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("51870.l1")
sage: E.isogeny_class()

## Elliptic curves in class 51870m

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
51870.l4 51870m1 [1, 1, 0, -313287, 62963829] 2 884736 $$\Gamma_0(N)$$-optimal
51870.l2 51870m2 [1, 1, 0, -4921287, 4200026229] 4 1769472
51870.l3 51870m3 [1, 1, 0, -4830087, 4363292469] 2 3538944
51870.l1 51870m4 [1, 1, 0, -78740487, 268900913589] 2 3538944

## Rank

sage: E.rank()

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

## Modular form None

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.