Elliptic curve isogeny class with Cremona label 458640dp (LMFDB label 458640.dp)

• Label: 458640dp
• Number of curves: $4$
• Conductor: $458640$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 458640dp have rank \(1\).

Complex multiplication

The elliptic curves in class 458640dp do not have complex multiplication.

Modular form 458640.2.a.dp

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.

Elliptic curves in class 458640dp

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
458640.dp3	458640dp1	\([0, 0, 0, -95403, -11001382]\)	\(273359449/9360\)	\(3288149575925760\)	\([2]\)	\(2359296\)	\(1.7493\)	\(\Gamma_0(N)\)-optimal^*
458640.dp2	458640dp2	\([0, 0, 0, -236523, 29104922]\)	\(4165509529/1368900\)	\(480891875479142400\)	\([2, 2]\)	\(4718592\)	\(2.0959\)	\(\Gamma_0(N)\)-optimal^*
458640.dp1	458640dp3	\([0, 0, 0, -3411723, 2425110842]\)	\(12501706118329/2570490\)	\(903008077288611840\)	\([2]\)	\(9437184\)	\(2.4424\)	\(\Gamma_0(N)\)-optimal^*
458640.dp4	458640dp4	\([0, 0, 0, 680757, 199902458]\)	\(99317171591/106616250\)	\(-37454078763279360000\)	\([2]\)	\(9437184\)	\(2.4424\)

^*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 458640dp1.