# Properties

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

Show commands: SageMath
sage: E = EllipticCurve("f1")

sage: E.isogeny_class()

## Elliptic curves in class 585.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
585.f1 585d1 $$[0, 0, 1, -42, 105]$$ $$-303464448/1625$$ $$-43875$$ $$$$ $$48$$ $$-0.26471$$ $$\Gamma_0(N)$$-optimal
585.f2 585d2 $$[0, 0, 1, 108, 560]$$ $$7077888/10985$$ $$-216217755$$ $$[]$$ $$144$$ $$0.28459$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form585.2.a.f

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. 