# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("2415.c1")

sage: E.isogeny_class()

## Elliptic curves in class 2415.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2415.c1 2415h3 [1, 0, 0, -4430, -113505]  2048
2415.c2 2415h2 [1, 0, 0, -405, 0] [2, 2] 1024
2415.c3 2415h1 [1, 0, 0, -280, 1775]  512 $$\Gamma_0(N)$$-optimal
2415.c4 2415h4 [1, 0, 0, 1620, 405]  2048

## Rank

sage: E.rank()

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

## Modular form2415.2.a.c

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} - q^{7} + 3q^{8} + q^{9} - q^{10} - q^{12} + 2q^{13} + q^{14} + q^{15} - q^{16} - 6q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

## 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 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. 