# Properties

 Label 1710.a Number of curves $2$ Conductor $1710$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 1710.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1710.a1 1710a2 $$[1, -1, 0, -4875, -128539]$$ $$651038076963/7220000$$ $$142111260000$$ $$$$ $$3840$$ $$0.95431$$
1710.a2 1710a1 $$[1, -1, 0, -555, 1925]$$ $$961504803/486400$$ $$9573811200$$ $$$$ $$1920$$ $$0.60774$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 1710.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 1710.a do not have complex multiplication.

## Modular form1710.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - q^{5} - 4 q^{7} - q^{8} + q^{10} + 6 q^{11} + 4 q^{14} + q^{16} - 4 q^{17} - q^{19} + O(q^{20})$$ ## 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. 