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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 18810a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
18810.e2 | 18810a1 | \([1, -1, 0, -570, 6580]\) | \(-759299343867/223646720\) | \(-6038461440\) | \([2]\) | \(16640\) | \(0.59129\) | \(\Gamma_0(N)\)-optimal |
18810.e1 | 18810a2 | \([1, -1, 0, -9690, 369556]\) | \(3726975084864507/222543200\) | \(6008666400\) | \([2]\) | \(33280\) | \(0.93786\) |
Rank
sage: E.rank()
The elliptic curves in class 18810a have rank \(1\).
Complex multiplication
The elliptic curves in class 18810a do not have complex multiplication.Modular form 18810.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.