Show commands:
SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 12240.cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12240.cf1 | 12240z4 | \([0, 0, 0, -242751387, 1455762872234]\) | \(1059623036730633329075378/154307373046875\) | \(230379673500000000000\) | \([4]\) | \(1720320\) | \(3.3157\) | |
| 12240.cf2 | 12240z3 | \([0, 0, 0, -28220907, -21762561094]\) | \(1664865424893526702418/826424127435466125\) | \(1233844610868131440896000\) | \([2]\) | \(1720320\) | \(3.3157\) | |
| 12240.cf3 | 12240z2 | \([0, 0, 0, -15215907, 22607897906]\) | \(521902963282042184836/6241849278890625\) | \(4659515519294736000000\) | \([2, 2]\) | \(860160\) | \(2.9691\) | |
| 12240.cf4 | 12240z1 | \([0, 0, 0, -182127, 908139854]\) | \(-3579968623693264/1906997690433375\) | \(-355891536979438176000\) | \([2]\) | \(430080\) | \(2.6225\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12240.cf have rank \(0\).
Complex multiplication
The elliptic curves in class 12240.cf do not have complex multiplication.Modular form 12240.2.a.cf
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
