Properties

 Label 100002a 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 100002a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
100002.a2 100002a1 [1, 1, 0, 112, 0] 2 75136 $$\Gamma_0(N)$$-optimal
100002.a1 100002a2 [1, 1, 0, -448, -560] 2 150272

Rank

sage: E.rank()

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

Modular form None

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 Cremona 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 Cremona labels. 