Elliptic curve isogeny class with Cremona label 493680ha (LMFDB label 493680.ha)

• Label: 493680ha
• Number of curves: $8$
• Conductor: $493680$
• CM: no
• Rank: $1$

Elliptic curves in class 493680ha

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
493680.ha6	493680ha1	\([0, 1, 0, -154920, 28784820]\)	\(-56667352321/16711680\)	\(-121265195141038080\)	\([2]\)	\(3932160\)	\(1.9925\)	\(\Gamma_0(N)\)-optimal^*
493680.ha5	493680ha2	\([0, 1, 0, -2633000, 1643501748]\)	\(278202094583041/16646400\)	\(120791502972518400\)	\([2, 2]\)	\(7864320\)	\(2.3391\)	\(\Gamma_0(N)\)-optimal^*
493680.ha2	493680ha3	\([0, 1, 0, -42127400, 105229414068]\)	\(1139466686381936641/4080\)	\(29605760532480\)	\([2]\)	\(15728640\)	\(2.6857\)	\(\Gamma_0(N)\)-optimal^*
493680.ha4	493680ha4	\([0, 1, 0, -2787880, 1439122100]\)	\(330240275458561/67652010000\)	\(490904217549250560000\)	\([2, 2]\)	\(15728640\)	\(2.6857\)
493680.ha7	493680ha5	\([0, 1, 0, 5924120, 8642203700]\)	\(3168685387909439/6278181696900\)	\(-45556456837701062246400\)	\([2]\)	\(31457280\)	\(3.0323\)
493680.ha3	493680ha6	\([0, 1, 0, -13977960, -18841778892]\)	\(41623544884956481/2962701562500\)	\(21498292399161600000000\)	\([2, 2]\)	\(31457280\)	\(3.0323\)
493680.ha8	493680ha7	\([0, 1, 0, 12680760, -82193561100]\)	\(31077313442863199/420227050781250\)	\(-3049299371250000000000000\)	\([2]\)	\(62914560\)	\(3.3788\)
493680.ha1	493680ha8	\([0, 1, 0, -219677960, -1253288618892]\)	\(161572377633716256481/914742821250\)	\(6637661008512906240000\)	\([2]\)	\(62914560\)	\(3.3788\)

^*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 493680ha1.

Rank

The elliptic curves in class 493680ha have rank \(1\).

Complex multiplication

The elliptic curves in class 493680ha do not have complex multiplication.

Modular form 493680.2.a.ha

Isogeny 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.