Properties

 Label 665.a Number of curves 2 Conductor 665 CM no Rank 1 Graph Related objects

Show commands for: SageMath
sage: E = EllipticCurve("665.a1")
sage: E.isogeny_class()

Elliptic curves in class 665.a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
665.a1 665d2 [0, -1, 1, -16660, -1081562] 1 3000
665.a2 665d1 [0, -1, 1, -210, 6798] 5 600 $$\Gamma_0(N)$$-optimal

Rank

sage: E.rank()

The elliptic curves in class 665.a have rank $$1$$.

Modular form665.2.a.a

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels. 