# Properties

 Label 585.d Number of curves $2$ Conductor $585$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 585.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
585.d1 585a2 $$[1, -1, 1, -3593, 83782]$$ $$260549802603/4225$$ $$83160675$$ $$$$ $$384$$ $$0.65208$$
585.d2 585a1 $$[1, -1, 1, -218, 1432]$$ $$-57960603/8125$$ $$-159924375$$ $$$$ $$192$$ $$0.30551$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 585.d have rank $$1$$.

## Complex multiplication

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

## Modular form585.2.a.d

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. 