Elliptic curve isogeny class with LMFDB label 405720.gr (Cremona label 405720gr)

• Label: 405720.gr
• Number of curves: $6$
• Conductor: $405720$
• CM: no
• Rank: $1$

Elliptic curves in class 405720.gr

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
405720.gr1	405720gr6	\([0, 0, 0, -627172707, -6045452429506]\)	\(155324313723954725282/13018359375\)	\(2286662011653600000000\)	\([2]\)	\(88080384\)	\(3.5409\)
405720.gr2	405720gr3	\([0, 0, 0, -53978547, 152482439774]\)	\(198048499826486404/242568272835\)	\(21303439194857085987840\)	\([2]\)	\(44040192\)	\(3.1943\)	\(\Gamma_0(N)\)-optimal^*
405720.gr3	405720gr4	\([0, 0, 0, -39284427, -94024215754]\)	\(76343005935514084/694180580625\)	\(60966067889903621760000\)	\([2, 2]\)	\(44040192\)	\(3.1943\)
405720.gr4	405720gr5	\([0, 0, 0, -11501427, -224443174354]\)	\(-957928673903042/123339801817575\)	\(-21664514799210818307225600\)	\([2]\)	\(88080384\)	\(3.5409\)
405720.gr5	405720gr2	\([0, 0, 0, -4277847, 1004646314]\)	\(394315384276816/208332909225\)	\(4574183808223581753600\)	\([2, 2]\)	\(22020096\)	\(2.8478\)	\(\Gamma_0(N)\)-optimal^*
405720.gr6	405720gr1	\([0, 0, 0, 1016358, 122631761]\)	\(84611246065664/53699121315\)	\(-73689045380735505840\)	\([2]\)	\(11010048\)	\(2.5012\)	\(\Gamma_0(N)\)-optimal^*

^*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 405720.gr1.

Rank

The elliptic curves in class 405720.gr have rank \(1\).

Complex multiplication

The elliptic curves in class 405720.gr do not have complex multiplication.

Modular form 405720.2.a.gr

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.

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

Isogeny graph

The vertices are labelled with LMFDB labels.