# Properties

 Label 578.a Number of curves 4 Conductor 578 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("578.a1")

sage: E.isogeny_class()

## Elliptic curves in class 578.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
578.a1 578a4 [1, 1, 1, -32663, -1583717]  3456
578.a2 578a3 [1, 1, 1, -29773, -1989473]  1728
578.a3 578a2 [1, 1, 1, -12433, 528295]  1152
578.a4 578a1 [1, 1, 1, -873, 5783]  576 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 578.a have rank $$0$$.

## Modular form578.2.a.a

sage: E.q_eigenform(10)

$$q + q^{2} + 2q^{3} + q^{4} + 2q^{6} + 4q^{7} + q^{8} + q^{9} - 6q^{11} + 2q^{12} + 2q^{13} + 4q^{14} + q^{16} + q^{18} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 