# Properties

 Label 1682.d Number of curves $2$ Conductor $1682$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("d1")

E.isogeny_class()

## Elliptic curves in class 1682.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1682.d1 1682a2 $$[1, 0, 1, -382673, -91764536]$$ $$-10418796526321/82044596$$ $$-48802039062823316$$ $$[]$$ $$16800$$ $$2.0308$$
1682.d2 1682a1 $$[1, 0, 1, 4187, 173584]$$ $$13651919/29696$$ $$-17663873340416$$ $$[]$$ $$3360$$ $$1.2261$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 1682.d have rank $$1$$.

## Complex multiplication

The elliptic curves in class 1682.d do not have complex multiplication.

## Modular form1682.2.a.d

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 2 q^{7} - q^{8} - 2 q^{9} - q^{10} + 3 q^{11} + q^{12} - q^{13} + 2 q^{14} + q^{15} + q^{16} - 8 q^{17} + 2 q^{18} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 