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SageMath
E = EllipticCurve("ha1")
E.isogeny_class()
Elliptic curves in class 109200.ha
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 109200.ha1 | 109200gl4 | \([0, 1, 0, -166808, -26277612]\) | \(8020417344913/187278\) | \(11985792000000\) | \([2]\) | \(589824\) | \(1.6218\) | |
| 109200.ha2 | 109200gl2 | \([0, 1, 0, -10808, -381612]\) | \(2181825073/298116\) | \(19079424000000\) | \([2, 2]\) | \(294912\) | \(1.2753\) | |
| 109200.ha3 | 109200gl1 | \([0, 1, 0, -2808, 50388]\) | \(38272753/4368\) | \(279552000000\) | \([2]\) | \(147456\) | \(0.92869\) | \(\Gamma_0(N)\)-optimal |
| 109200.ha4 | 109200gl3 | \([0, 1, 0, 17192, -2005612]\) | \(8780064047/32388174\) | \(-2072843136000000\) | \([2]\) | \(589824\) | \(1.6218\) |
Rank
sage: E.rank()
The elliptic curves in class 109200.ha have rank \(1\).
Complex multiplication
The elliptic curves in class 109200.ha do not have complex multiplication.Modular form 109200.2.a.ha
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.
