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

• Label: 458640ld
• Number of curves: $8$
• Conductor: $458640$
• CM: no
• Rank: $1$

Elliptic curves in class 458640ld

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
458640.ld7	458640ld1	\([0, 0, 0, -181515747, -936235683614]\)	\(1882742462388824401/11650189824000\)	\(4092688753123953475584000\)	\([2]\)	\(84934656\)	\(3.5607\)	\(\Gamma_0(N)\)-optimal^*
458640.ld6	458640ld2	\([0, 0, 0, -292153827, 339930313954]\)	\(7850236389974007121/4400862921000000\)	\(1546014481559140110336000000\)	\([2, 2]\)	\(169869312\)	\(3.9073\)	\(\Gamma_0(N)\)-optimal^*
458640.ld5	458640ld3	\([0, 0, 0, -1121374947, 13823907590626]\)	\(443915739051786565201/21894701746029840\)	\(7691565626199681134799421440\)	\([2]\)	\(254803968\)	\(4.1100\)	\(\Gamma_0(N)\)-optimal^*
458640.ld4	458640ld4	\([0, 0, 0, -3502633827, 79657407097954]\)	\(13527956825588849127121/25701087819771000\)	\(9028741561475505267879936000\)	\([2]\)	\(339738624\)	\(4.2539\)	\(\Gamma_0(N)\)-optimal^*
458640.ld8	458640ld5	\([0, 0, 0, 1148116893, 2697077374306]\)	\(476437916651992691759/284661685546875000\)	\(-100001089809109944000000000000\)	\([2]\)	\(339738624\)	\(4.2539\)
458640.ld2	458640ld6	\([0, 0, 0, -17724001827, 908217379192354]\)	\(1752803993935029634719121/4599740941532100\)	\(1615879938703753273278873600\)	\([2, 2]\)	\(509607936\)	\(4.4566\)	\(\Gamma_0(N)\)-optimal^*
458640.ld1	458640ld7	\([0, 0, 0, -283583850627, 58125989146260994]\)	\(7179471593960193209684686321/49441793310\)	\(17368804669368117288960\)	\([2]\)	\(1019215872\)	\(4.8032\)	\(\Gamma_0(N)\)-optimal^*
458640.ld3	458640ld8	\([0, 0, 0, -17506183107, 931627794634306]\)	\(-1688971789881664420008241/89901485966373558750\)	\(-31582215059340478754623933440000\)	\([2]\)	\(1019215872\)	\(4.8032\)

^*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 6 curves highlighted, and conditionally curve 458640ld1.

Rank

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

Complex multiplication

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

Modular form 458640.2.a.ld

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.