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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 4410.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4410.a1 | 4410b1 | \([1, -1, 0, -251085, -48258715]\) | \(551105805571803/1376829440\) | \(4373530383237120\) | \([2]\) | \(53760\) | \(1.8786\) | \(\Gamma_0(N)\)-optimal |
4410.a2 | 4410b2 | \([1, -1, 0, -157005, -84931099]\) | \(-134745327251163/903920796800\) | \(-2871325201213526400\) | \([2]\) | \(107520\) | \(2.2251\) |
Rank
sage: E.rank()
The elliptic curves in class 4410.a have rank \(0\).
Complex multiplication
The elliptic curves in class 4410.a do not have complex multiplication.Modular form 4410.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.