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

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

Elliptic curves in class 405042.z

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
405042.z1	405042z3	\([1, 1, 0, -506933174, 4392920075220]\)	\(306234591284035366263793/1727485056\)	\(81271056373854336\)	\([2]\)	\(74317824\)	\(3.3145\)	\(\Gamma_0(N)\)-optimal^*
405042.z2	405042z2	\([1, 1, 0, -31683894, 68626876500]\)	\(74768347616680342513/5615307472896\)	\(264177087148275941376\)	\([2, 2]\)	\(37158912\)	\(2.9679\)	\(\Gamma_0(N)\)-optimal^*
405042.z3	405042z4	\([1, 1, 0, -29604534, 78025999572]\)	\(-60992553706117024753/20624795251201152\)	\(-970311663037374504054912\)	\([2]\)	\(74317824\)	\(3.3145\)
405042.z4	405042z1	\([1, 1, 0, -2110774, 922175572]\)	\(22106889268753393/4969545596928\)	\(233796650777148653568\)	\([2]\)	\(18579456\)	\(2.6213\)	\(\Gamma_0(N)\)-optimal^*

^*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 405042.z1.

Rank

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

Complex multiplication

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

Modular form 405042.2.a.z

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.