# Properties

 Label 585.i Number of curves $2$ Conductor $585$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 585.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
585.i1 585c2 $$[1, -1, 0, -399, -2970]$$ $$260549802603/4225$$ $$114075$$ $$$$ $$128$$ $$0.10278$$
585.i2 585c1 $$[1, -1, 0, -24, -45]$$ $$-57960603/8125$$ $$-219375$$ $$$$ $$64$$ $$-0.24380$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 585.i have rank $$0$$.

## Complex multiplication

The elliptic curves in class 585.i do not have complex multiplication.

## Modular form585.2.a.i

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} + q^{5} + 2 q^{7} - 3 q^{8} + q^{10} + 4 q^{11} - q^{13} + 2 q^{14} - q^{16} + 4 q^{17} + 6 q^{19} + O(q^{20})$$

## 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. 