# Properties

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

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

sage: E.isogeny_class()

## Elliptic curves in class 2415.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2415.c1 2415h3 $$[1, 0, 0, -4430, -113505]$$ $$9614816895690721/34652610405$$ $$34652610405$$ $$$$ $$2048$$ $$0.88359$$
2415.c2 2415h2 $$[1, 0, 0, -405, 0]$$ $$7347774183121/4251692025$$ $$4251692025$$ $$[2, 2]$$ $$1024$$ $$0.53702$$
2415.c3 2415h1 $$[1, 0, 0, -280, 1775]$$ $$2428257525121/8150625$$ $$8150625$$ $$$$ $$512$$ $$0.19044$$ $$\Gamma_0(N)$$-optimal
2415.c4 2415h4 $$[1, 0, 0, 1620, 405]$$ $$470166844956479/272118787605$$ $$-272118787605$$ $$$$ $$2048$$ $$0.88359$$

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 2415.c do not have complex multiplication.

## Modular form2415.2.a.c

sage: E.q_eigenform(10)

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

## 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 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. 