# Properties

 Label 1002.d Number of curves $2$ Conductor $1002$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 1002.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1002.d1 1002d2 $$[1, 0, 1, -125, -544]$$ $$213525509833/669336$$ $$669336$$ $$$$ $$288$$ $$-0.014852$$
1002.d2 1002d1 $$[1, 0, 1, -5, -16]$$ $$-10218313/96192$$ $$-96192$$ $$$$ $$144$$ $$-0.36143$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 1002.d have rank $$1$$.

## Complex multiplication

The elliptic curves in class 1002.d do not have complex multiplication.

## Modular form1002.2.a.d

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} + 2 q^{5} - q^{6} - 4 q^{7} - q^{8} + q^{9} - 2 q^{10} - 4 q^{11} + q^{12} + 4 q^{14} + 2 q^{15} + q^{16} - 4 q^{17} - q^{18} - 4 q^{19} + O(q^{20})$$

## 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. 