Elliptic curve isogeny class with LMFDB label 412269.d (Cremona label 412269d)

• Label: 412269.d
• Number of curves: $2$
• Conductor: $412269$
• CM: no
• Rank: $1$

Elliptic curves in class 412269.d

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
412269.d1	412269d2	\([1, 0, 0, -12513, -406290]\)	\(244140625/61347\)	\(54445688318307\)	\([2]\)	\(948480\)	\(1.3456\)	\(\Gamma_0(N)\)-optimal^*
412269.d2	412269d1	\([1, 0, 0, 1902, -40149]\)	\(857375/1287\)	\(-1142217237447\)	\([2]\)	\(474240\)	\(0.99905\)	\(\Gamma_0(N)\)-optimal^*

^*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 412269.d1.

Rank

The elliptic curves in class 412269.d have rank \(1\).

Complex multiplication

The elliptic curves in class 412269.d do not have complex multiplication.

Modular form 412269.2.a.d

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.

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

Isogeny graph

The vertices are labelled with LMFDB labels.