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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 100023.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
100023.c1 | 100023b1 | \([1, 0, 0, -343, -1312]\) | \(4463599344625/1871530353\) | \(1871530353\) | \([2]\) | \(43008\) | \(0.47616\) | \(\Gamma_0(N)\)-optimal |
100023.c2 | 100023b2 | \([1, 0, 0, 1142, -9331]\) | \(164701765793375/133697843433\) | \(-133697843433\) | \([2]\) | \(86016\) | \(0.82273\) |
Rank
sage: E.rank()
The elliptic curves in class 100023.c have rank \(2\).
Complex multiplication
The elliptic curves in class 100023.c do not have complex multiplication.Modular form 100023.2.a.c
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.