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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 356928.cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
356928.cd1 | 356928cd2 | \([0, -1, 0, -140833, -15254591]\) | \(244140625/61347\) | \(77623525827674112\) | \([2]\) | \(2752512\) | \(1.9508\) | |
356928.cd2 | 356928cd1 | \([0, -1, 0, 21407, -1529087]\) | \(857375/1287\) | \(-1628465576804352\) | \([2]\) | \(1376256\) | \(1.6043\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 356928.cd have rank \(2\).
Complex multiplication
The elliptic curves in class 356928.cd do not have complex multiplication.Modular form 356928.2.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 LMFDB 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 LMFDB labels.