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

• Label: 400710b
• Number of curves: $4$
• Conductor: $400710$
• CM: no
• Rank: $0$

Elliptic curves in class 400710b

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
400710.b3	400710b1	\([1, 1, 0, -19517833, -73826099963]\)	\(-17478209248027211809/39921724625688000\)	\(-1878152706054887191128000\)	\([2]\)	\(74649600\)	\(3.3462\)	\(\Gamma_0(N)\)-optimal^*
400710.b2	400710b2	\([1, 1, 0, -409007953, -3181256175347]\)	\(160841222880596489520289/158621068526625000\)	\(7462467913996445081625000\)	\([2]\)	\(149299200\)	\(3.6927\)	\(\Gamma_0(N)\)-optimal^*
400710.b4	400710b3	\([1, 1, 0, 169379027, 1624758910633]\)	\(11423021746642244003231/30848851120710937500\)	\(-1451311378811683401023437500\)	\([2]\)	\(223948800\)	\(3.8955\)	\(\Gamma_0(N)\)-optimal^*
400710.b1	400710b4	\([1, 1, 0, -1476336943, 18437722403347]\)	\(7564122771096983025656449/1255581642150878906250\)	\(59069944522414833068847656250\)	\([2]\)	\(447897600\)	\(4.2421\)	\(\Gamma_0(N)\)-optimal^*

^*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 400710b1.

Rank

The elliptic curves in class 400710b have rank \(0\).

Complex multiplication

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

Modular form 400710.2.a.b

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.