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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 333270h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333270.h3 | 333270h1 | \([1, -1, 0, -3599415, -2627500275]\) | \(47788676405569/579600\) | \(62549367321747600\) | \([2]\) | \(8650752\) | \(2.3715\) | \(\Gamma_0(N)\)-optimal |
333270.h2 | 333270h2 | \([1, -1, 0, -3694635, -2481070959]\) | \(51682540549249/5249002500\) | \(566462707807576702500\) | \([2, 2]\) | \(17301504\) | \(2.7180\) | |
333270.h1 | 333270h3 | \([1, -1, 0, -13549905, 16482439575]\) | \(2549399737314529/388286718750\) | \(41903189436248880468750\) | \([2]\) | \(34603008\) | \(3.0646\) | |
333270.h4 | 333270h4 | \([1, -1, 0, 4637115, -12074247909]\) | \(102181603702751/642612880350\) | \(-69349601619562440358350\) | \([2]\) | \(34603008\) | \(3.0646\) |
Rank
sage: E.rank()
The elliptic curves in class 333270h have rank \(1\).
Complex multiplication
The elliptic curves in class 333270h do not have complex multiplication.Modular form 333270.2.a.h
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.