Elliptic curve isogeny class with Cremona label 474810ba (LMFDB label 474810.ba)

• Label: 474810ba
• Number of curves: $8$
• Conductor: $474810$
• CM: no
• Rank: $1$

Elliptic curves in class 474810ba

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
474810.ba7	474810ba1	\([1, 1, 0, -558868202, -4983249420204]\)	\(164083032511008797673646489/3779535863669623787520\)	\(444658614824867568977940480\)	\([2]\)	\(238878720\)	\(3.8995\)	\(\Gamma_0(N)\)-optimal^*
474810.ba4	474810ba2	\([1, 1, 0, -8891863082, -322732010387436]\)	\(660866552951225193140994678169/363054521201227329600\)	\(42713001364803194100110400\)	\([2, 2]\)	\(477757440\)	\(4.2461\)
474810.ba6	474810ba3	\([1, 1, 0, -5450323337, 152878168412229]\)	\(152195662006675487969752714249/2254051004206282702848000\)	\(265186846593864953707364352000\)	\([2]\)	\(716636160\)	\(4.4488\)	\(\Gamma_0(N)\)-optimal^*
474810.ba5	474810ba4	\([1, 1, 0, -8841853682, -326541516439956]\)	\(-649778658927959232413187423769/15498405515425377751317720\)	\(-1823371910484280267064778440280\)	\([2]\)	\(955514880\)	\(4.5927\)
474810.ba2	474810ba5	\([1, 1, 0, -142269790562, -20654676546340164]\)	\(2706908330196708836642873424493849/816939805815000\)	\(96112151214328935000\)	\([2]\)	\(955514880\)	\(4.5927\)
474810.ba3	474810ba6	\([1, 1, 0, -10652571017, -185895321857979]\)	\(1136315122909965387044499819529/530704359775758422016000000\)	\(62436837223258202591760384000000\)	\([2, 2]\)	\(1433272320\)	\(4.7954\)
474810.ba8	474810ba7	\([1, 1, 0, 37751588983, -1405496218049979]\)	\(50575615882668425252678113940471/36522079745400816582633408000\)	\(-4296786159966660670130237817792000\)	\([2]\)	\(2866544640\)	\(5.1420\)
474810.ba1	474810ba8	\([1, 1, 0, -142292693897, -20647693758005691]\)	\(2708215857449597952771459256806409/1815677562935478375000000000\)	\(213612649601796095340375000000000\)	\([2]\)	\(2866544640\)	\(5.1420\)

^*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 474810ba1.

Rank

The elliptic curves in class 474810ba have rank \(1\).

Complex multiplication

The elliptic curves in class 474810ba do not have complex multiplication.

Modular form 474810.2.a.ba

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.