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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 48960cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
48960.cn2 | 48960cd1 | \([0, 0, 0, 229812, 13163888]\) | \(7023836099951/4456448000\) | \(-851640475189248000\) | \([]\) | \(483840\) | \(2.1295\) | \(\Gamma_0(N)\)-optimal |
48960.cn1 | 48960cd2 | \([0, 0, 0, -3825228, 2972458352]\) | \(-32391289681150609/1228250000000\) | \(-234722230272000000000\) | \([]\) | \(1451520\) | \(2.6788\) |
Rank
sage: E.rank()
The elliptic curves in class 48960cd have rank \(1\).
Complex multiplication
The elliptic curves in class 48960cd do not have complex multiplication.Modular form 48960.2.a.cd
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.