Show commands:
SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 10890.ba
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10890.ba1 | 10890ba7 | \([1, -1, 0, -5808204, -5386341240]\) | \(16778985534208729/81000\) | \(104608905489000\) | \([2]\) | \(276480\) | \(2.3124\) | |
| 10890.ba2 | 10890ba8 | \([1, -1, 0, -493884, -18059112]\) | \(10316097499609/5859375000\) | \(7567195130859375000\) | \([2]\) | \(276480\) | \(2.3124\) | |
| 10890.ba3 | 10890ba6 | \([1, -1, 0, -363204, -84000240]\) | \(4102915888729/9000000\) | \(11623211721000000\) | \([2, 2]\) | \(138240\) | \(1.9658\) | |
| 10890.ba4 | 10890ba5 | \([1, -1, 0, -314199, 67866255]\) | \(2656166199049/33750\) | \(43587043953750\) | \([2]\) | \(92160\) | \(1.7631\) | |
| 10890.ba5 | 10890ba4 | \([1, -1, 0, -74619, -6738957]\) | \(35578826569/5314410\) | \(6863390289133290\) | \([2]\) | \(92160\) | \(1.7631\) | |
| 10890.ba6 | 10890ba2 | \([1, -1, 0, -20169, 1003833]\) | \(702595369/72900\) | \(94148014940100\) | \([2, 2]\) | \(46080\) | \(1.4165\) | |
| 10890.ba7 | 10890ba3 | \([1, -1, 0, -14724, -2246832]\) | \(-273359449/1536000\) | \(-1983694800384000\) | \([2]\) | \(69120\) | \(1.6193\) | |
| 10890.ba8 | 10890ba1 | \([1, -1, 0, 1611, 76005]\) | \(357911/2160\) | \(-2789570813040\) | \([2]\) | \(23040\) | \(1.0700\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 10890.ba have rank \(0\).
Complex multiplication
The elliptic curves in class 10890.ba do not have complex multiplication.Modular form 10890.2.a.ba
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
