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SageMath
E = EllipticCurve("eg1")
E.isogeny_class()
Elliptic curves in class 89280eg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 89280.r4 | 89280eg1 | \([0, 0, 0, 800052, -309032048]\) | \(296354077829711/387386634240\) | \(-74030738665887498240\) | \([2]\) | \(2211840\) | \(2.4985\) | \(\Gamma_0(N)\)-optimal |
| 89280.r3 | 89280eg2 | \([0, 0, 0, -4913868, -3012858992]\) | \(68663623745397169/19216056254400\) | \(3672245537199253094400\) | \([2]\) | \(4423680\) | \(2.8450\) | |
| 89280.r2 | 89280eg3 | \([0, 0, 0, -22838988, -42262604912]\) | \(-6894246873502147249/47925198774000\) | \(-9158648111102951424000\) | \([2]\) | \(6635520\) | \(3.0478\) | |
| 89280.r1 | 89280eg4 | \([0, 0, 0, -366031308, -2695413792368]\) | \(28379906689597370652529/1357352437500\) | \(259394090287104000000\) | \([2]\) | \(13271040\) | \(3.3943\) |
Rank
sage: E.rank()
The elliptic curves in class 89280eg have rank \(1\).
Complex multiplication
The elliptic curves in class 89280eg do not have complex multiplication.Modular form 89280.2.a.eg
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.
