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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 3648.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3648.j1 | 3648y1 | \([0, -1, 0, -6113, 185985]\) | \(96386901625/18468\) | \(4841275392\) | \([2]\) | \(3840\) | \(0.85903\) | \(\Gamma_0(N)\)-optimal |
3648.j2 | 3648y2 | \([0, -1, 0, -5473, 225793]\) | \(-69173457625/42633378\) | \(-11176084242432\) | \([2]\) | \(7680\) | \(1.2056\) |
Rank
sage: E.rank()
The elliptic curves in class 3648.j have rank \(1\).
Complex multiplication
The elliptic curves in class 3648.j do not have complex multiplication.Modular form 3648.2.a.j
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.