Elliptic curve isogeny class with Cremona label 417690cy (LMFDB label 417690.cy)

• Label: 417690cy
• Number of curves: $3$
• Conductor: $417690$
• CM: no
• Rank: $1$

Elliptic curves in class 417690cy

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
417690.cy2	417690cy1	\([1, -1, 1, -1088882, -1143475111]\)	\(-5288128867919358659043/17870267558905307000\)	\(-482497224090443289000\)	\([3]\)	\(28553472\)	\(2.6550\)	\(\Gamma_0(N)\)-optimal^*
417690.cy3	417690cy2	\([1, -1, 1, 9555583, 26802839441]\)	\(397090089854833871209893/1507166867843000000000\)	\(-366241548885849000000000\)	\([9]\)	\(85660416\)	\(3.2043\)	\(\Gamma_0(N)\)-optimal^*
417690.cy1	417690cy3	\([1, -1, 1, -123351932, -527281210871]\)	\(-10545575477636568394902267/18237629526473630\)	\(-358971261969580459290\)	\([]\)	\(85660416\)	\(3.2043\)

^*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 417690cy1.

Rank

The elliptic curves in class 417690cy have rank \(1\).

Complex multiplication

The elliptic curves in class 417690cy do not have complex multiplication.

Modular form 417690.2.a.cy

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.

\(\left(\begin{array}{rrr} 1 & 3 & 3 \\ 3 & 1 & 9 \\ 3 & 9 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.