Elliptic curve isogeny class with Cremona label 486720hr (LMFDB label 486720.hr)

• Label: 486720hr
• Number of curves: $4$
• Conductor: $486720$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 486720hr have rank \(1\).

Complex multiplication

The elliptic curves in class 486720hr do not have complex multiplication.

Modular form 486720.2.a.hr

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 486720hr

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
486720.hr4	486720hr1	\([0, 0, 0, 1052532, -324417808]\)	\(3774555693/3515200\)	\(-120091934173062758400\)	\([2]\)	\(18579456\)	\(2.5401\)	\(\Gamma_0(N)\)-optimal^*
486720.hr3	486720hr2	\([0, 0, 0, -5437068, -2922853648]\)	\(520300455507/193072360\)	\(6596049484455472005120\)	\([2]\)	\(37158912\)	\(2.8867\)	\(\Gamma_0(N)\)-optimal^*
486720.hr2	486720hr3	\([0, 0, 0, -24256908, -46489961232]\)	\(-63378025803/812500\)	\(-20235535320858624000000\)	\([2]\)	\(55738368\)	\(3.0894\)
486720.hr1	486720hr4	\([0, 0, 0, -389296908, -2956442825232]\)	\(261984288445803/42250\)	\(1052247836684648448000\)	\([2]\)	\(111476736\)	\(3.4360\)

^*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 486720hr1.