Show commands:
SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 159120z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 159120.ek7 | 159120z1 | \([0, 0, 0, -301845027, 1943629784354]\) | \(1018563973439611524445729/42904970360310988800\) | \(128113555016362847580979200\) | \([2]\) | \(53084160\) | \(3.7742\) | \(\Gamma_0(N)\)-optimal |
| 159120.ek6 | 159120z2 | \([0, 0, 0, -800246307, -6128776387294]\) | \(18980483520595353274840609/5549773448629762560000\) | \(16571534721233292927959040000\) | \([2, 2]\) | \(106168320\) | \(4.1208\) | |
| 159120.ek5 | 159120z3 | \([0, 0, 0, -3714161187, -86556842855134]\) | \(1897660325010178513043539489/14258428094958372000000\) | \(42575438156696179458048000000\) | \([2]\) | \(159252480\) | \(4.3235\) | |
| 159120.ek8 | 159120z4 | \([0, 0, 0, 2142837213, -40735318265566]\) | \(364421318680576777174674911/450962301637624725000000\) | \(-1346566217293121226854400000000\) | \([2]\) | \(212336640\) | \(4.4673\) | |
| 159120.ek4 | 159120z5 | \([0, 0, 0, -11717750307, -488156229494494]\) | \(59589391972023341137821784609/8834417507562311995200\) | \(26379429326900942620675276800\) | \([2]\) | \(212336640\) | \(4.4673\) | |
| 159120.ek2 | 159120z6 | \([0, 0, 0, -59319000867, -5560819946447326]\) | \(7730680381889320597382223137569/441370202660156250000\) | \(1317924363219984000000000000\) | \([2, 2]\) | \(318504960\) | \(4.6701\) | |
| 159120.ek3 | 159120z7 | \([0, 0, 0, -59211435747, -5581991709347614]\) | \(-7688701694683937879808871873249/58423707246780395507812500\) | \(-174452255059570312500000000000000\) | \([2]\) | \(637009920\) | \(5.0166\) | |
| 159120.ek1 | 159120z8 | \([0, 0, 0, -949104000867, -355892486813447326]\) | \(31664865542564944883878115208137569/103216295812500\) | \(308202207835392000000\) | \([2]\) | \(637009920\) | \(5.0166\) |
Rank
sage: E.rank()
The elliptic curves in class 159120z have rank \(1\).
Complex multiplication
The elliptic curves in class 159120z do not have complex multiplication.Modular form 159120.2.a.z
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().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
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
