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SageMath
E = EllipticCurve("pj1")
E.isogeny_class()
Elliptic curves in class 446400.pj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
446400.pj1 | 446400pj2 | \([0, 0, 0, -539172300, 4495296922000]\) | \(5805223604235668521/435937500000000\) | \(1301702400000000000000000000\) | \([2]\) | \(198180864\) | \(3.9481\) | \(\Gamma_0(N)\)-optimal* |
446400.pj2 | 446400pj1 | \([0, 0, 0, 32219700, 311564698000]\) | \(1238798620042199/14760960000000\) | \(-44075990384640000000000000\) | \([2]\) | \(99090432\) | \(3.6016\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 446400.pj have rank \(1\).
Complex multiplication
The elliptic curves in class 446400.pj do not have complex multiplication.Modular form 446400.2.a.pj
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.