Elliptic curve isogeny class with Cremona label 428400gk (LMFDB label 428400.gk)

• Label: 428400gk
• Number of curves: $6$
• Conductor: $428400$
• CM: no
• Rank: $0$

Elliptic curves in class 428400gk

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
428400.gk5	428400gk1	\([0, 0, 0, 125925, 42570250]\)	\(4733169839/19518975\)	\(-910677297600000000\)	\([2]\)	\(6291456\)	\(2.1287\)	\(\Gamma_0(N)\)-optimal^*
428400.gk4	428400gk2	\([0, 0, 0, -1332075, 522252250]\)	\(5602762882081/716900625\)	\(33447715560000000000\)	\([2, 2]\)	\(12582912\)	\(2.4753\)	\(\Gamma_0(N)\)-optimal^*
428400.gk2	428400gk3	\([0, 0, 0, -20610075, 36013050250]\)	\(20751759537944401/418359375\)	\(19518975000000000000\)	\([2]\)	\(25165824\)	\(2.8218\)	\(\Gamma_0(N)\)-optimal^*
428400.gk3	428400gk4	\([0, 0, 0, -5382075, -4268897750]\)	\(369543396484081/45120132225\)	\(2105124889089600000000\)	\([2, 2]\)	\(25165824\)	\(2.8218\)
428400.gk6	428400gk5	\([0, 0, 0, 7847925, -21957407750]\)	\(1145725929069119/5127181719135\)	\(-239213790287962560000000\)	\([4]\)	\(50331648\)	\(3.1684\)
428400.gk1	428400gk6	\([0, 0, 0, -83412075, -293213987750]\)	\(1375634265228629281/24990412335\)	\(1165952677901760000000\)	\([2]\)	\(50331648\)	\(3.1684\)

^*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 428400gk1.

Rank

The elliptic curves in class 428400gk have rank \(0\).

Complex multiplication

The elliptic curves in class 428400gk do not have complex multiplication.

Modular form 428400.2.a.gk

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.

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

Isogeny graph

The vertices are labelled with Cremona labels.