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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 67600.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
67600.r1 | 67600ca1 | \([0, 1, 0, -4008, -104012]\) | \(-658489/40\) | \(-432640000000\) | \([]\) | \(82944\) | \(0.98765\) | \(\Gamma_0(N)\)-optimal |
67600.r2 | 67600ca2 | \([0, 1, 0, 21992, -156012]\) | \(108750551/64000\) | \(-692224000000000\) | \([]\) | \(248832\) | \(1.5370\) |
Rank
sage: E.rank()
The elliptic curves in class 67600.r have rank \(0\).
Complex multiplication
The elliptic curves in class 67600.r do not have complex multiplication.Modular form 67600.2.a.r
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.