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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 140a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
140.a2 | 140a1 | \([0, 1, 0, -5, -25]\) | \(-65536/875\) | \(-224000\) | \([3]\) | \(12\) | \(-0.29176\) | \(\Gamma_0(N)\)-optimal |
140.a1 | 140a2 | \([0, 1, 0, -805, -9065]\) | \(-225637236736/1715\) | \(-439040\) | \([]\) | \(36\) | \(0.25755\) |
Rank
sage: E.rank()
The elliptic curves in class 140a have rank \(0\).
Complex multiplication
The elliptic curves in class 140a do not have complex multiplication.Modular form 140.2.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 Cremona 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 Cremona labels.