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SageMath
E = EllipticCurve("ex1")
E.isogeny_class()
Elliptic curves in class 158400ex
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
158400.me2 | 158400ex1 | \([0, 0, 0, -69600, 8215000]\) | \(-3196715008/649539\) | \(-7576222896000000\) | \([2]\) | \(983040\) | \(1.7694\) | \(\Gamma_0(N)\)-optimal |
158400.me1 | 158400ex2 | \([0, 0, 0, -1163100, 482794000]\) | \(932410994128/29403\) | \(5487305472000000\) | \([2]\) | \(1966080\) | \(2.1160\) |
Rank
sage: E.rank()
The elliptic curves in class 158400ex have rank \(0\).
Complex multiplication
The elliptic curves in class 158400ex do not have complex multiplication.Modular form 158400.2.a.ex
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.