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SageMath
E = EllipticCurve("ej1")
E.isogeny_class()
Elliptic curves in class 100800ej
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 100800.f4 | 100800ej1 | \([0, 0, 0, 6000, 61000]\) | \(2048000/1323\) | \(-15431472000000\) | \([2]\) | \(221184\) | \(1.2195\) | \(\Gamma_0(N)\)-optimal |
| 100800.f3 | 100800ej2 | \([0, 0, 0, -25500, 502000]\) | \(9826000/5103\) | \(952342272000000\) | \([2]\) | \(442368\) | \(1.5661\) | |
| 100800.f2 | 100800ej3 | \([0, 0, 0, -102000, 12913000]\) | \(-10061824000/352947\) | \(-4116773808000000\) | \([2]\) | \(663552\) | \(1.7688\) | |
| 100800.f1 | 100800ej4 | \([0, 0, 0, -1645500, 812446000]\) | \(2640279346000/3087\) | \(576108288000000\) | \([2]\) | \(1327104\) | \(2.1154\) |
Rank
sage: E.rank()
The elliptic curves in class 100800ej have rank \(0\).
Complex multiplication
The elliptic curves in class 100800ej do not have complex multiplication.Modular form 100800.2.a.ej
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
