Elliptic curve isogeny class with Cremona label 476850ib (LMFDB label 476850.ib)

• Label: 476850ib
• Number of curves: $8$
• Conductor: $476850$
• CM: no
• Rank: $1$

Elliptic curves in class 476850ib

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
476850.ib8	476850ib1	\([1, 0, 0, 313366162, -903423075708]\)	\(9023321954633914439/6156756739584000\)	\(-2322017822155059864000000000\)	\([4]\)	\(382205952\)	\(3.9397\)	\(\Gamma_0(N)\)-optimal^*
476850.ib7	476850ib2	\([1, 0, 0, -1372081838, -7539031851708]\)	\(757443433548897303481/373234243041000000\)	\(140765113977576677015625000000\)	\([2, 2]\)	\(764411904\)	\(4.2862\)	\(\Gamma_0(N)\)-optimal^*
476850.ib6	476850ib3	\([1, 0, 0, -5641298213, -167063826227583]\)	\(-52643812360427830814761/1504091705903677440\)	\(-567267458337151899402240000000\)	\([4]\)	\(1146617856\)	\(4.4890\)
476850.ib5	476850ib4	\([1, 0, 0, -11757874838, 485484947651292]\)	\(476646772170172569823801/5862293314453125000\)	\(2210961083997671905517578125000\)	\([2]\)	\(1528823808\)	\(4.6328\)	\(\Gamma_0(N)\)-optimal^*
476850.ib4	476850ib5	\([1, 0, 0, -17953456838, -925235231226708]\)	\(1696892787277117093383481/1440538624914939000\)	\(543298444625772800676421875000\)	\([2]\)	\(1528823808\)	\(4.6328\)
476850.ib3	476850ib6	\([1, 0, 0, -90870866213, -10543508041523583]\)	\(220031146443748723000125481/172266701724057600\)	\(64970303113544676249600000000\)	\([2, 2]\)	\(2293235712\)	\(4.8355\)
476850.ib2	476850ib7	\([1, 0, 0, -91481234213, -10394687505395583]\)	\(224494757451893010998773801/6152490825146276160000\)	\(2320408934591174623516485000000000\)	\([2]\)	\(4586471424\)	\(5.1821\)
476850.ib1	476850ib8	\([1, 0, 0, -1453933586213, -674784417438323583]\)	\(901247067798311192691198986281/552431869440\)	\(208349411975109240000000\)	\([2]\)	\(4586471424\)	\(5.1821\)

^*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 476850ib1.

Rank

The elliptic curves in class 476850ib have rank \(1\).

Complex multiplication

The elliptic curves in class 476850ib do not have complex multiplication.

Modular form 476850.2.a.ib

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.