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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 457776cr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
457776.cr2 | 457776cr1 | \([0, 0, 0, -39015, 2387718]\) | \(54000/11\) | \(1337880954085632\) | \([2]\) | \(1769472\) | \(1.6181\) | \(\Gamma_0(N)\)-optimal |
457776.cr1 | 457776cr2 | \([0, 0, 0, -195075, -31040334]\) | \(1687500/121\) | \(58866761979767808\) | \([2]\) | \(3538944\) | \(1.9647\) |
Rank
sage: E.rank()
The elliptic curves in class 457776cr have rank \(1\).
Complex multiplication
The elliptic curves in class 457776cr do not have complex multiplication.Modular form 457776.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.