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SageMath
E = EllipticCurve("cw1")
E.isogeny_class()
Elliptic curves in class 68970cw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 68970.cw4 | 68970cw1 | \([1, 0, 0, 1510, 19812]\) | \(214921799/218880\) | \(-387759271680\) | \([2]\) | \(163840\) | \(0.91049\) | \(\Gamma_0(N)\)-optimal |
| 68970.cw3 | 68970cw2 | \([1, 0, 0, -8170, 180500]\) | \(34043726521/11696400\) | \(20720886080400\) | \([2, 2]\) | \(327680\) | \(1.2571\) | |
| 68970.cw2 | 68970cw3 | \([1, 0, 0, -54150, -4720968]\) | \(9912050027641/311647500\) | \(552102556747500\) | \([2]\) | \(655360\) | \(1.6036\) | |
| 68970.cw1 | 68970cw4 | \([1, 0, 0, -117070, 15404720]\) | \(100162392144121/23457780\) | \(41556888194580\) | \([2]\) | \(655360\) | \(1.6036\) |
Rank
sage: E.rank()
The elliptic curves in class 68970cw have rank \(0\).
Complex multiplication
The elliptic curves in class 68970cw do not have complex multiplication.Modular form 68970.2.a.cw
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
