Elliptic curve isogeny class with Cremona label 459186bs (LMFDB label 459186.bs)

• Label: 459186bs
• Number of curves: $4$
• Conductor: $459186$
• CM: no
• Rank: $1$

Elliptic curves in class 459186bs

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
459186.bs3	459186bs1	\([1, 1, 1, -47534, -3898645]\)	\(19968681097/628992\)	\(374139110322432\)	\([2]\)	\(2408448\)	\(1.5711\)	\(\Gamma_0(N)\)-optimal^*
459186.bs2	459186bs2	\([1, 1, 1, -114814, 9503531]\)	\(281397674377/96589584\)	\(57453737128888464\)	\([2, 2]\)	\(4816896\)	\(1.9177\)	\(\Gamma_0(N)\)-optimal^*
459186.bs1	459186bs3	\([1, 1, 1, -1645434, 811548411]\)	\(828279937799497/193444524\)	\(115065314194944204\)	\([2]\)	\(9633792\)	\(2.2643\)	\(\Gamma_0(N)\)-optimal^*
459186.bs4	459186bs4	\([1, 1, 1, 339326, 66543515]\)	\(7264187703863/7406095788\)	\(-4405318492262271948\)	\([2]\)	\(9633792\)	\(2.2643\)

^*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 459186bs1.

Rank

The elliptic curves in class 459186bs have rank \(1\).

Complex multiplication

The elliptic curves in class 459186bs do not have complex multiplication.

Modular form 459186.2.a.bs

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.