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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 103488cf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
103488.e2 | 103488cf1 | \([0, -1, 0, -1270145, -860083839]\) | \(-7347774183121/6119866368\) | \(-188742672928953335808\) | \([2]\) | \(6193152\) | \(2.5884\) | \(\Gamma_0(N)\)-optimal |
103488.e1 | 103488cf2 | \([0, -1, 0, -23347585, -43403310719]\) | \(45637459887836881/13417633152\) | \(413812948388976525312\) | \([2]\) | \(12386304\) | \(2.9350\) |
Rank
sage: E.rank()
The elliptic curves in class 103488cf have rank \(1\).
Complex multiplication
The elliptic curves in class 103488cf do not have complex multiplication.Modular form 103488.2.a.cf
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.