Properties

 Label 493680cp Number of curves 2 Conductor 493680 CM no Rank 0 Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("493680.cp1")

sage: E.isogeny_class()

Elliptic curves in class 493680cp

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
493680.cp2 493680cp1 [0, -1, 0, -1703720, 377535600] [u'2'] 15482880 $$\Gamma_0(N)$$-optimal*
493680.cp1 493680cp2 [0, -1, 0, -22767400, 41797155952] [u'2'] 30965760 $$\Gamma_0(N)$$-optimal*
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 493680cp1.

Rank

sage: E.rank()

The elliptic curves in class 493680cp have rank $$0$$.

Modular form 493680.2.a.cp

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} - 2q^{7} + q^{9} + 4q^{13} - q^{15} - q^{17} - 6q^{19} + O(q^{20})$$

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.