# Properties

 Label 2415i Number of curves $2$ Conductor $2415$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 2415i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2415.d1 2415i1 $$[1, 0, 0, -230, -1173]$$ $$1345938541921/203765625$$ $$203765625$$ $$$$ $$768$$ $$0.31835$$ $$\Gamma_0(N)$$-optimal
2415.d2 2415i2 $$[1, 0, 0, 395, -6298]$$ $$6814692748079/21258460125$$ $$-21258460125$$ $$$$ $$1536$$ $$0.66492$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form2415.2.a.i

sage: E.q_eigenform(10)

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