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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 2178b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2178.c2 | 2178b1 | \([1, -1, 0, -38682, -1293836]\) | \(3723875/1728\) | \(2970335001724992\) | \([2]\) | \(12672\) | \(1.6635\) | \(\Gamma_0(N)\)-optimal |
2178.c1 | 2178b2 | \([1, -1, 0, -517842, -143221028]\) | \(8934171875/5832\) | \(10024880630821848\) | \([2]\) | \(25344\) | \(2.0101\) |
Rank
sage: E.rank()
The elliptic curves in class 2178b have rank \(0\).
Complex multiplication
The elliptic curves in class 2178b do not have complex multiplication.Modular form 2178.2.a.b
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.