Elliptic curve isogeny class with LMFDB label 431970.bz (Cremona label 431970bz)

• Label: 431970.bz
• Number of curves: $4$
• Conductor: $431970$
• CM: no
• Rank: $1$

Elliptic curves in class 431970.bz

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
431970.bz1	431970bz4	\([1, 0, 1, -102111419, -396776314618]\)	\(66464620505913166201729/74880071980801920\)	\(132654615198381430197120\)	\([2]\)	\(91750400\)	\(3.3507\)
431970.bz2	431970bz3	\([1, 0, 1, -72529339, 235730497286]\)	\(23818189767728437646209/232359312482640000\)	\(411638695981058201040000\)	\([2]\)	\(91750400\)	\(3.3507\)	\(\Gamma_0(N)\)-optimal^*
431970.bz3	431970bz2	\([1, 0, 1, -8021819, -2766705658]\)	\(32224493437735955329/16782725759385600\)	\(29731622429022912921600\)	\([2, 2]\)	\(45875200\)	\(3.0041\)	\(\Gamma_0(N)\)-optimal^*
431970.bz4	431970bz1	\([1, 0, 1, 1890501, -336204794]\)	\(421792317902132351/271682182840320\)	\(-481301559514780139520\)	\([2]\)	\(22937600\)	\(2.6575\)	\(\Gamma_0(N)\)-optimal^*

^*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 431970.bz1.

Rank

The elliptic curves in class 431970.bz have rank \(1\).

Complex multiplication

The elliptic curves in class 431970.bz do not have complex multiplication.

Modular form 431970.2.a.bz

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.

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

Isogeny graph

The vertices are labelled with LMFDB labels.