Elliptic curve isogeny class with LMFDB label 400710.e (Cremona label 400710e)

• Label: 400710.e
• Number of curves: $4$
• Conductor: $400710$
• CM: no
• Rank: $1$

Elliptic curves in class 400710.e

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
400710.e1	400710e4	\([1, 1, 0, -78573823, -266177262107]\)	\(1140343700217211191169/9496149787277280\)	\(446754732850422228883680\)	\([2]\)	\(59904000\)	\(3.3639\)
400710.e2	400710e2	\([1, 1, 0, -8395423, 2507760133]\)	\(1391008986004445569/747073209369600\)	\(35146717306290286617600\)	\([2, 2]\)	\(29952000\)	\(3.0173\)
400710.e3	400710e1	\([1, 1, 0, -6547103, 6437658117]\)	\(659704930833045889/895635947520\)	\(42135982206348165120\)	\([2]\)	\(14976000\)	\(2.6707\)	\(\Gamma_0(N)\)-optimal^*
400710.e4	400710e3	\([1, 1, 0, 32209857, 19716277797]\)	\(78554030949152410751/49024188678060000\)	\(-2306386146669558070860000\)	\([2]\)	\(59904000\)	\(3.3639\)

^*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 400710.e1.

Rank

The elliptic curves in class 400710.e have rank \(1\).

Complex multiplication

The elliptic curves in class 400710.e do not have complex multiplication.

Modular form 400710.2.a.e

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.

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

Isogeny graph

The vertices are labelled with LMFDB labels.