# Properties

 Label 2415.a Number of curves 4 Conductor 2415 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("2415.a1")

sage: E.isogeny_class()

## Elliptic curves in class 2415.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2415.a1 2415a3 [1, 1, 1, -5051, 136064]  1920
2415.a2 2415a2 [1, 1, 1, -326, 1874] [2, 2] 960
2415.a3 2415a1 [1, 1, 1, -81, -282]  480 $$\Gamma_0(N)$$-optimal
2415.a4 2415a4 [1, 1, 1, 479, 10568]  1920

## Rank

sage: E.rank()

The elliptic curves in class 2415.a have rank $$1$$.

## Modular form2415.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} - q^{4} - q^{5} + q^{6} - q^{7} + 3q^{8} + q^{9} + q^{10} + 4q^{11} + q^{12} - 6q^{13} + q^{14} + q^{15} - q^{16} - 2q^{17} - q^{18} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 