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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 67600cr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
67600.bc2 | 67600cr1 | \([0, -1, 0, 1192, 48112]\) | \(1331/8\) | \(-1124864000000\) | \([]\) | \(62208\) | \(0.99429\) | \(\Gamma_0(N)\)-optimal |
67600.bc1 | 67600cr2 | \([0, -1, 0, -128808, -18047888]\) | \(-1680914269/32768\) | \(-4607442944000000\) | \([]\) | \(311040\) | \(1.7990\) |
Rank
sage: E.rank()
The elliptic curves in class 67600cr have rank \(1\).
Complex multiplication
The elliptic curves in class 67600cr do not have complex multiplication.Modular form 67600.2.a.cr
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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.