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SageMath
E = EllipticCurve("ew1")
E.isogeny_class()
Elliptic curves in class 85680ew
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 85680.eb2 | 85680ew1 | \([0, 0, 0, 11733, 584026]\) | \(59822347031/83966400\) | \(-250722326937600\) | \([2]\) | \(221184\) | \(1.4486\) | \(\Gamma_0(N)\)-optimal |
| 85680.eb1 | 85680ew2 | \([0, 0, 0, -74667, 5785306]\) | \(15417797707369/4080067320\) | \(12183015736442880\) | \([2]\) | \(442368\) | \(1.7951\) |
Rank
sage: E.rank()
The elliptic curves in class 85680ew have rank \(0\).
Complex multiplication
The elliptic curves in class 85680ew do not have complex multiplication.Modular form 85680.2.a.ew
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
