# Properties

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

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

E.isogeny_class()

## Elliptic curves in class 70.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
70.a1 70a3 $$[1, -1, 1, -268, -1619]$$ $$2121328796049/120050$$ $$120050$$ $$$$ $$16$$ $$0.038988$$
70.a2 70a4 $$[1, -1, 1, -88, 317]$$ $$74565301329/5468750$$ $$5468750$$ $$$$ $$16$$ $$0.038988$$
70.a3 70a2 $$[1, -1, 1, -18, -19]$$ $$611960049/122500$$ $$122500$$ $$[2, 2]$$ $$8$$ $$-0.30759$$
70.a4 70a1 $$[1, -1, 1, 2, -3]$$ $$1367631/2800$$ $$-2800$$ $$$$ $$4$$ $$-0.65416$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 70.a do not have complex multiplication.

## Modular form70.2.a.a

sage: E.q_eigenform(10)

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

## 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. 