Elliptic curve isogeny class with LMFDB label 429429.bi (Cremona label 429429bi)

• Label: 429429.bi
• Number of curves: $6$
• Conductor: $429429$
• CM: no
• Rank: $1$

Elliptic curves in class 429429.bi

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
429429.bi1	429429bi5	\([1, 0, 0, -16032442, 24707234993]\)	\(53297461115137/147\)	\(1256995027090803\)	\([2]\)	\(11796480\)	\(2.5556\)	\(\Gamma_0(N)\)-optimal^*
429429.bi2	429429bi3	\([1, 0, 0, -1002427, 385664720]\)	\(13027640977/21609\)	\(184778268982348041\)	\([2, 2]\)	\(5898240\)	\(2.2091\)	\(\Gamma_0(N)\)-optimal^*
429429.bi3	429429bi4	\([1, 0, 0, -797937, -272752182]\)	\(6570725617/45927\)	\(392721160606798023\)	\([2]\)	\(5898240\)	\(2.2091\)
429429.bi4	429429bi6	\([1, 0, 0, -695692, 626206307]\)	\(-4354703137/17294403\)	\(-147884207942205882147\)	\([2]\)	\(11796480\)	\(2.5556\)
429429.bi5	429429bi2	\([1, 0, 0, -82222, 1939235]\)	\(7189057/3969\)	\(33938865731451681\)	\([2, 2]\)	\(2949120\)	\(1.8625\)	\(\Gamma_0(N)\)-optimal^*
429429.bi6	429429bi1	\([1, 0, 0, 20023, 241968]\)	\(103823/63\)	\(-538712154467487\)	\([2]\)	\(1474560\)	\(1.5159\)	\(\Gamma_0(N)\)-optimal^*

^*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 429429.bi1.

Rank

The elliptic curves in class 429429.bi have rank \(1\).

Complex multiplication

The elliptic curves in class 429429.bi do not have complex multiplication.

Modular form 429429.2.a.bi

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 LMFDB numbering.

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

Isogeny graph

The vertices are labelled with LMFDB labels.