# Properties

 Label 70.a Number of curves $4$ Conductor $70$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("70.a1")

sage: E.isogeny_class()

## Elliptic curves in class 70.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
70.a1 70a3 [1, -1, 1, -268, -1619]  16
70.a2 70a4 [1, -1, 1, -88, 317]  16
70.a3 70a2 [1, -1, 1, -18, -19] [2, 2] 8
70.a4 70a1 [1, -1, 1, 2, -3]  4 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 70.a have rank $$0$$.

## Modular form70.2.a.a

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - 3q^{9} - q^{10} + 4q^{11} - 6q^{13} - q^{14} + q^{16} + 2q^{17} - 3q^{18} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 