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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 3450.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3450.r1 | 3450q2 | \([1, 1, 1, -2668, -35059]\) | \(84013940106985/28705554432\) | \(717638860800\) | \([]\) | \(5184\) | \(0.97696\) | |
3450.r2 | 3450q1 | \([1, 1, 1, -1093, 13451]\) | \(5776556465785/1073088\) | \(26827200\) | \([]\) | \(1728\) | \(0.42765\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3450.r have rank \(1\).
Complex multiplication
The elliptic curves in class 3450.r do not have complex multiplication.Modular form 3450.2.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.