# Properties

 Label 10002.a Number of curves $2$ Conductor $10002$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("10002.a1")

sage: E.isogeny_class()

## Elliptic curves in class 10002.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
10002.a1 10002c1 [1, 1, 0, -101, -435]  1536 $$\Gamma_0(N)$$-optimal
10002.a2 10002c2 [1, 1, 0, -61, -731]  3072

## Rank

sage: E.rank()

The elliptic curves in class 10002.a have rank $$0$$.

## Modular form 10002.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} - 2q^{5} + q^{6} - 2q^{7} - q^{8} + q^{9} + 2q^{10} - q^{12} - 2q^{13} + 2q^{14} + 2q^{15} + q^{16} + 6q^{17} - q^{18} + 4q^{19} + O(q^{20})$$

## 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. 