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SageMath
E = EllipticCurve("ex1")
E.isogeny_class()
Elliptic curves in class 381150ex
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
381150.ex1 | 381150ex1 | \([1, -1, 0, -569729067, 5233681854341]\) | \(37537160298467283/5519360000\) | \(3007157336457120000000000\) | \([2]\) | \(123863040\) | \(3.7112\) | \(\Gamma_0(N)\)-optimal |
381150.ex2 | 381150ex2 | \([1, -1, 0, -517457067, 6232861134341]\) | \(-28124139978713043/14526050000000\) | \(-7914344747804627343750000000\) | \([2]\) | \(247726080\) | \(4.0577\) |
Rank
sage: E.rank()
The elliptic curves in class 381150ex have rank \(1\).
Complex multiplication
The elliptic curves in class 381150ex do not have complex multiplication.Modular form 381150.2.a.ex
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.