Elliptic curve isogeny class with Cremona label 459186r (LMFDB label 459186.r)

• Label: 459186r
• Number of curves: $4$
• Conductor: $459186$
• CM: no
• Rank: $0$

Elliptic curves in class 459186r

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
459186.r3	459186r1	\([1, 1, 0, -5904, -158928]\)	\(38272753/4368\)	\(2598188266128\)	\([2]\)	\(1161216\)	\(1.1145\)	\(\Gamma_0(N)\)-optimal^*
459186.r2	459186r2	\([1, 1, 0, -22724, 1142940]\)	\(2181825073/298116\)	\(177326349163236\)	\([2, 2]\)	\(2322432\)	\(1.4610\)	\(\Gamma_0(N)\)-optimal^*
459186.r1	459186r3	\([1, 1, 0, -350714, 79794942]\)	\(8020417344913/187278\)	\(111397321910238\)	\([2]\)	\(4644864\)	\(1.8076\)	\(\Gamma_0(N)\)-optimal^*
459186.r4	459186r4	\([1, 1, 0, 36146, 6146890]\)	\(8780064047/32388174\)	\(-19265241219805854\)	\([2]\)	\(4644864\)	\(1.8076\)

^*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 459186r1.

Rank

The elliptic curves in class 459186r have rank \(0\).

Complex multiplication

The elliptic curves in class 459186r do not have complex multiplication.

Modular form 459186.2.a.r

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.