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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 18259.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
18259.a1 | 18259a3 | \([0, -1, 1, -739329, 244930132]\) | \(-50357871050752/19\) | \(-16862569939\) | \([]\) | \(90720\) | \(1.7504\) | |
18259.a2 | 18259a2 | \([0, -1, 1, -8969, 350827]\) | \(-89915392/6859\) | \(-6087387747979\) | \([]\) | \(30240\) | \(1.2011\) | |
18259.a3 | 18259a1 | \([0, -1, 1, 641, 62]\) | \(32768/19\) | \(-16862569939\) | \([]\) | \(10080\) | \(0.65182\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 18259.a have rank \(1\).
Complex multiplication
The elliptic curves in class 18259.a do not have complex multiplication.Modular form 18259.2.a.a
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.