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

• Label: 431970bq
• Number of curves: $8$
• Conductor: $431970$
• CM: no
• Rank: $0$

Elliptic curves in class 431970bq

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
431970.bq7	431970bq1	\([1, 0, 1, -3911449, -3549402628]\)	\(-3735772816268612449/909650165760000\)	\(-1611500757303951360000\)	\([2]\)	\(26542080\)	\(2.7877\)	\(\Gamma_0(N)\)-optimal^*
431970.bq6	431970bq2	\([1, 0, 1, -65863449, -205735949828]\)	\(17836145204788591940449/770635366502400\)	\(1365227560516358246400\)	\([2, 2]\)	\(53084160\)	\(3.1343\)	\(\Gamma_0(N)\)-optimal^*
431970.bq8	431970bq3	\([1, 0, 1, 28148711, 24146672636]\)	\(1392333139184610040991/947901937500000000\)	\(-1679266104299437500000000\)	\([2]\)	\(79626240\)	\(3.3370\)	\(\Gamma_0(N)\)-optimal^*
431970.bq5	431970bq4	\([1, 0, 1, -69154649, -184039042948]\)	\(20645800966247918737249/3688936444974392640\)	\(6535175937395279999711040\)	\([2]\)	\(106168320\)	\(3.4809\)	\(\Gamma_0(N)\)-optimal^*
431970.bq3	431970bq5	\([1, 0, 1, -1053804249, -13167124069508]\)	\(73054578035931991395831649/136386452160\)	\(241616919575021760\)	\([2]\)	\(106168320\)	\(3.4809\)
431970.bq4	431970bq6	\([1, 0, 1, -123101289, 201351172636]\)	\(116454264690812369959009/57505157319440250000\)	\(101873894005984888730250000\)	\([2, 2]\)	\(159252480\)	\(3.6836\)	\(\Gamma_0(N)\)-optimal^*
431970.bq1	431970bq7	\([1, 0, 1, -1609283789, 24829773089636]\)	\(260174968233082037895439009/223081361502731896500\)	\(395202239865141221295436500\)	\([2]\)	\(318504960\)	\(4.0302\)	\(\Gamma_0(N)\)-optimal^*
431970.bq2	431970bq8	\([1, 0, 1, -1056918789, -13085377744364]\)	\(73704237235978088924479009/899277423164136103500\)	\(1593124811058080119652563500\)	\([2]\)	\(318504960\)	\(4.0302\)

^*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 6 curves highlighted, and conditionally curve 431970bq1.

Rank

The elliptic curves in class 431970bq have rank \(0\).

Complex multiplication

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

Modular form 431970.2.a.bq

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.