# Properties

 Label 51870n Number of curves 4 Conductor 51870 CM no Rank 2 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("51870.k1")

sage: E.isogeny_class()

## Elliptic curves in class 51870n

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
51870.k3 51870n1 [1, 1, 0, -1092, -13104]  49152 $$\Gamma_0(N)$$-optimal
51870.k2 51870n2 [1, 1, 0, -3972, 80784] [2, 2] 98304
51870.k4 51870n3 [1, 1, 0, 6948, 458616]  196608
51870.k1 51870n4 [1, 1, 0, -60972, 5769384]  196608

## Rank

sage: E.rank()

The elliptic curves in class 51870n have rank $$2$$.

## Modular form 51870.2.a.k

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. 