Elliptic curve isogeny class with Cremona label 417450bd (LMFDB label 417450.bd)

• Label: 417450bd
• Number of curves: $4$
• Conductor: $417450$
• CM: no
• Rank: $2$

Elliptic curves in class 417450bd

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
417450.bd3	417450bd1	\([1, 1, 0, -37875, -1561875]\)	\(217081801/88320\)	\(2444754180000000\)	\([2]\)	\(1966080\)	\(1.6504\)	\(\Gamma_0(N)\)-optimal^*
417450.bd2	417450bd2	\([1, 1, 0, -279875, 55792125]\)	\(87587538121/1904400\)	\(52715012006250000\)	\([2, 2]\)	\(3932160\)	\(1.9970\)	\(\Gamma_0(N)\)-optimal^*
417450.bd1	417450bd3	\([1, 1, 0, -4454375, 3616640625]\)	\(353108405631241/172500\)	\(4774910507812500\)	\([2]\)	\(7864320\)	\(2.3436\)	\(\Gamma_0(N)\)-optimal^*
417450.bd4	417450bd4	\([1, 1, 0, 22625, 170439625]\)	\(46268279/453342420\)	\(-12548808608087812500\)	\([2]\)	\(7864320\)	\(2.3436\)

^*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 417450bd1.

Rank

The elliptic curves in class 417450bd have rank \(2\).

Complex multiplication

The elliptic curves in class 417450bd do not have complex multiplication.

Modular form 417450.2.a.bd

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.