# Properties

 Label 504.a Number of curves $2$ Conductor $504$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 504.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
504.a1 504a2 [0, 0, 0, -111, 450]  64
504.a2 504a1 [0, 0, 0, -6, 9]  32 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 504.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 504.a do not have complex multiplication.

## Modular form504.2.a.a

sage: E.q_eigenform(10)

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

## 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. 