Elliptic curve isogeny class with Cremona label 428736fh (LMFDB label 428736.fh)

• Label: 428736fh
• Number of curves: $4$
• Conductor: $428736$
• CM: no
• Rank: $2$

Elliptic curves in class 428736fh

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
428736.fh3	428736fh1	\([0, 1, 0, -9889, -310849]\)	\(408023180713/80247321\)	\(21036353716224\)	\([2]\)	\(786432\)	\(1.2730\)	\(\Gamma_0(N)\)-optimal^*
428736.fh2	428736fh2	\([0, 1, 0, -48609, 3832191]\)	\(48455467135993/3635004681\)	\(952894667096064\)	\([2, 2]\)	\(1572864\)	\(1.6196\)	\(\Gamma_0(N)\)-optimal^*
428736.fh1	428736fh3	\([0, 1, 0, -763169, 256357695]\)	\(187519537050946633/1186707753\)	\(311088317202432\)	\([2]\)	\(3145728\)	\(1.9662\)	\(\Gamma_0(N)\)-optimal^*
428736.fh4	428736fh4	\([0, 1, 0, 46431, 17080767]\)	\(42227808999767/504359959257\)	\(-132214937159467008\)	\([2]\)	\(3145728\)	\(1.9662\)

^*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 428736fh1.

Rank

The elliptic curves in class 428736fh have rank \(2\).

Complex multiplication

The elliptic curves in class 428736fh do not have complex multiplication.

Modular form 428736.2.a.fh

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.