Elliptic curve isogeny class with Cremona label 414960p (LMFDB label 414960.p)

• Label: 414960p
• Number of curves: $8$
• Conductor: $414960$
• CM: no
• Rank: $1$

Elliptic curves in class 414960p

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
414960.p7	414960p1	\([0, -1, 0, -5161416, -4431010320]\)	\(3712533999213317890249/76090919904090000\)	\(311668407927152640000\)	\([2]\)	\(15925248\)	\(2.7230\)	\(\Gamma_0(N)\)-optimal^*
414960.p6	414960p2	\([0, -1, 0, -10993416, 7428944880]\)	\(35872512095393194378249/14944558319037792900\)	\(61212910874778799718400\)	\([2, 2]\)	\(31850496\)	\(3.0695\)	\(\Gamma_0(N)\)-optimal^*
414960.p5	414960p3	\([0, -1, 0, -48208056, 126975167856]\)	\(3024980849878413455018809/50557689000000000000\)	\(207084294144000000000000\)	\([2]\)	\(47775744\)	\(3.2723\)	\(\Gamma_0(N)\)-optimal^*
414960.p4	414960p4	\([0, -1, 0, -151760616, 719373135600]\)	\(94371532824107026279203049/40995077600666342790\)	\(167915837852329340067840\)	\([2]\)	\(63700992\)	\(3.4161\)	\(\Gamma_0(N)\)-optimal^*
414960.p8	414960p5	\([0, -1, 0, 36461784, 54428574960]\)	\(1308812680909424992398551/1070002284841633041990\)	\(-4382729358711328939991040\)	\([2]\)	\(63700992\)	\(3.4161\)
414960.p2	414960p6	\([0, -1, 0, -768208056, 8195583167856]\)	\(12240533203187013248735018809/3506282465049000000\)	\(14361732976840704000000\)	\([2, 2]\)	\(95551488\)	\(3.6188\)	\(\Gamma_0(N)\)-optimal^*
414960.p1	414960p7	\([0, -1, 0, -12291328056, 524505107135856]\)	\(50137213659805457275731367898809/4113897879000\)	\(16850525712384000\)	\([2]\)	\(191102976\)	\(3.9654\)	\(\Gamma_0(N)\)-optimal^*
414960.p3	414960p8	\([0, -1, 0, -765088056, 8265451199856]\)	\(-12091997009671629064982138809/207252595706436249879000\)	\(-848906632013562879504384000\)	\([2]\)	\(191102976\)	\(3.9654\)

^*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 414960p1.

Rank

The elliptic curves in class 414960p have rank \(1\).

Complex multiplication

The elliptic curves in class 414960p do not have complex multiplication.

Modular form 414960.2.a.p

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.