# Properties

 Label 100002.a Number of curves 2 Conductor 100002 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("100002.a1")

sage: E.isogeny_class()

## Elliptic curves in class 100002.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
100002.a1 100002a2 [1, 1, 0, -448, -560]  150272
100002.a2 100002a1 [1, 1, 0, 112, 0]  75136 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 100002.a have rank $$1$$.

## Modular form 100002.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + 4q^{5} + q^{6} - q^{7} - q^{8} + q^{9} - 4q^{10} - 6q^{11} - q^{12} + 4q^{13} + q^{14} - 4q^{15} + q^{16} - 4q^{17} - q^{18} + 4q^{19} + O(q^{20})$$

## 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. 