Elliptic curve isogeny class with Cremona label 405600by (LMFDB label 405600.by)

• Label: 405600by
• Number of curves: $4$
• Conductor: $405600$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 405600by have rank \(1\).

Complex multiplication

The elliptic curves in class 405600by do not have complex multiplication.

Modular form 405600.2.a.by

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.

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

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 405600by

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
405600.by3	405600by1	\([0, -1, 0, -6766955758, -214241224047488]\)	\(7099759044484031233216/577161945398025\)	\(2785850472504695652225000000\)	\([2, 2]\)	\(433520640\)	\(4.3113\)	\(\Gamma_0(N)\)-optimal^*
405600.by2	405600by2	\([0, -1, 0, -7231072008, -183170497574988]\)	\(1082883335268084577352/251301565117746585\)	\(9703877249795402209578120000000\)	\([2]\)	\(867041280\)	\(4.6579\)	\(\Gamma_0(N)\)-optimal^*
405600.by4	405600by3	\([0, -1, 0, -6304952008, -244750103682488]\)	\(-717825640026599866952/254764560814329735\)	\(-9837599000157232750924920000000\)	\([2]\)	\(867041280\)	\(4.6579\)
405600.by1	405600by4	\([0, -1, 0, -108269179633, -13712108457168863]\)	\(454357982636417669333824/3003024375\)	\(927681605150040000000000\)	\([2]\)	\(867041280\)	\(4.6579\)

^*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 405600by1.