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SageMath
E = EllipticCurve("yd1")
E.isogeny_class()
Elliptic curves in class 235200yd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
235200.yd1 | 235200yd1 | \([0, 1, 0, -2770133, -1775465637]\) | \(1248870793216/42525\) | \(80048379600000000\) | \([2]\) | \(4423680\) | \(2.3351\) | \(\Gamma_0(N)\)-optimal |
235200.yd2 | 235200yd2 | \([0, 1, 0, -2647633, -1939493137]\) | \(-68150496976/14467005\) | \(-435719339838720000000\) | \([2]\) | \(8847360\) | \(2.6817\) |
Rank
sage: E.rank()
The elliptic curves in class 235200yd have rank \(1\).
Complex multiplication
The elliptic curves in class 235200yd do not have complex multiplication.Modular form 235200.2.a.yd
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.