# Properties

 Label 285.a Number of curves $2$ Conductor $285$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 285.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
285.a1 285a2 $$[1, 0, 0, -76, -19]$$ $$48587168449/28048275$$ $$28048275$$ $$$$ $$80$$ $$0.11864$$
285.a2 285a1 $$[1, 0, 0, 19, 0]$$ $$756058031/438615$$ $$-438615$$ $$$$ $$40$$ $$-0.22793$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 285.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 285.a do not have complex multiplication.

## Modular form285.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} - 2q^{7} + 3q^{8} + q^{9} + q^{10} - 6q^{11} - q^{12} + 2q^{14} - q^{15} - q^{16} - 6q^{17} - q^{18} + q^{19} + O(q^{20})$$ ## 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. 