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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 650.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
650.j1 | 650h3 | \([1, 1, 1, -11488, -478719]\) | \(-10730978619193/6656\) | \(-104000000\) | \([]\) | \(648\) | \(0.85911\) | |
650.j2 | 650h2 | \([1, 1, 1, -113, -969]\) | \(-10218313/17576\) | \(-274625000\) | \([]\) | \(216\) | \(0.30980\) | |
650.j3 | 650h1 | \([1, 1, 1, 12, 31]\) | \(12167/26\) | \(-406250\) | \([]\) | \(72\) | \(-0.23951\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 650.j have rank \(0\).
Complex multiplication
The elliptic curves in class 650.j do not have complex multiplication.Modular form 650.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.