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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 8001.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8001.c1 | 8001g2 | \([0, 0, 1, -107544, 13574592]\) | \(188692544389906432/301112301\) | \(219510867429\) | \([3]\) | \(24192\) | \(1.4411\) | |
8001.c2 | 8001g1 | \([0, 0, 1, -1704, 7227]\) | \(750593769472/403418421\) | \(294092028909\) | \([]\) | \(8064\) | \(0.89175\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8001.c have rank \(2\).
Complex multiplication
The elliptic curves in class 8001.c do not have complex multiplication.Modular form 8001.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.