# Properties

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

Show commands: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 585.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
585.e1 585b2 $$[0, 0, 1, -378, -2842]$$ $$-303464448/1625$$ $$-31984875$$ $$[]$$ $$144$$ $$0.28459$$
585.e2 585b1 $$[0, 0, 1, 12, -21]$$ $$7077888/10985$$ $$-296595$$ $$$$ $$48$$ $$-0.26471$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form585.2.a.e

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 