# Properties

 Label 54.a Number of curves $3$ Conductor $54$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("54.a1")

sage: E.isogeny_class()

## Elliptic curves in class 54.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
54.a1 54a2 [1, -1, 0, -123, -667] [] 18
54.a2 54a3 [1, -1, 0, -3, 3]  18
54.a3 54a1 [1, -1, 0, 12, 8]  6 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 54.a have rank $$0$$.

## Modular form54.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + 3q^{5} - q^{7} - q^{8} - 3q^{10} - 3q^{11} - 4q^{13} + q^{14} + q^{16} + 2q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 