# Properties

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

Show commands: 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 j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
504.a1 504a2 $$[0, 0, 0, -111, 450]$$ $$21882096/7$$ $$48384$$ $$$$ $$64$$ $$-0.12630$$
504.a2 504a1 $$[0, 0, 0, -6, 9]$$ $$-55296/49$$ $$-21168$$ $$$$ $$32$$ $$-0.47287$$ $$\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)

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

## 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.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 