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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 42432.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 42432.e1 | 42432bp4 | \([0, -1, 0, -117729, 15587073]\) | \(2753580869496292/39328497\) | \(2577432379392\) | \([2]\) | \(163840\) | \(1.5209\) | |
| 42432.e2 | 42432bp2 | \([0, -1, 0, -7569, 230769]\) | \(2927363579728/320445801\) | \(5250184003584\) | \([2, 2]\) | \(81920\) | \(1.1744\) | |
| 42432.e3 | 42432bp1 | \([0, -1, 0, -1789, -24707]\) | \(618724784128/87947613\) | \(90058355712\) | \([2]\) | \(40960\) | \(0.82778\) | \(\Gamma_0(N)\)-optimal |
| 42432.e4 | 42432bp3 | \([0, -1, 0, 10111, 1132449]\) | \(1744147297148/9513325341\) | \(-623465289547776\) | \([2]\) | \(163840\) | \(1.5209\) |
Rank
sage: E.rank()
The elliptic curves in class 42432.e have rank \(1\).
Complex multiplication
The elliptic curves in class 42432.e do not have complex multiplication.Modular form 42432.2.a.e
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 LMFDB 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 LMFDB labels.
