Elliptic curve isogeny class with Cremona label 405042bt (LMFDB label 405042.bt)

• Label: 405042bt
• Number of curves: $4$
• Conductor: $405042$
• CM: no
• Rank: $1$

Elliptic curves in class 405042bt

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
405042.bt3	405042bt1	\([1, 0, 1, -139715, -20111506]\)	\(6411014266033/296208\)	\(13935366319248\)	\([2]\)	\(2654208\)	\(1.5973\)	\(\Gamma_0(N)\)-optimal^*
405042.bt2	405042bt2	\([1, 0, 1, -146935, -17919514]\)	\(7457162887153/1370924676\)	\(64496359167059556\)	\([2, 2]\)	\(5308416\)	\(1.9439\)	\(\Gamma_0(N)\)-optimal^*
405042.bt1	405042bt3	\([1, 0, 1, -699265, 208535786]\)	\(803760366578833/65593817586\)	\(3085918936486663266\)	\([2]\)	\(10616832\)	\(2.2905\)	\(\Gamma_0(N)\)-optimal^*
405042.bt4	405042bt4	\([1, 0, 1, 289875, -104058446]\)	\(57258048889007/132611470002\)	\(-6238823436949161762\)	\([2]\)	\(10616832\)	\(2.2905\)

^*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 405042bt1.

Rank

The elliptic curves in class 405042bt have rank \(1\).

Complex multiplication

The elliptic curves in class 405042bt do not have complex multiplication.

Modular form 405042.2.a.bt

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.