# Properties

 Label 2415.i Number of curves $2$ Conductor $2415$ CM no Rank $1$ Graph

# Learn more

Show commands: SageMath
sage: E = EllipticCurve("i1")

sage: E.isogeny_class()

## Elliptic curves in class 2415.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2415.i1 2415g1 $$[1, 0, 1, -1438, -21037]$$ $$328523283207001/1109390625$$ $$1109390625$$ $$[2]$$ $$1152$$ $$0.59964$$ $$\Gamma_0(N)$$-optimal
2415.i2 2415g2 $$[1, 0, 1, -813, -39287]$$ $$-59323563117001/630142750125$$ $$-630142750125$$ $$[2]$$ $$2304$$ $$0.94621$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form2415.2.a.i

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} - q^{7} - 3 q^{8} + q^{9} + q^{10} - 2 q^{11} - q^{12} - q^{14} + q^{15} - q^{16} - 2 q^{17} + q^{18} - 4 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.