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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 41650e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
41650.k2 | 41650e1 | \([1, 1, 0, 67350, 17852500]\) | \(375078431/1740800\) | \(-156802587200000000\) | \([]\) | \(387072\) | \(1.9819\) | \(\Gamma_0(N)\)-optimal |
41650.k1 | 41650e2 | \([1, 1, 0, -618650, -539865500]\) | \(-290707016929/1228250000\) | \(-110634637941406250000\) | \([]\) | \(1161216\) | \(2.5312\) |
Rank
sage: E.rank()
The elliptic curves in class 41650e have rank \(0\).
Complex multiplication
The elliptic curves in class 41650e do not have complex multiplication.Modular form 41650.2.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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.