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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 22320bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22320.q2 | 22320bf1 | \([0, 0, 0, 357, -37942]\) | \(1685159/209250\) | \(-624817152000\) | \([]\) | \(23040\) | \(0.94275\) | \(\Gamma_0(N)\)-optimal |
22320.q1 | 22320bf2 | \([0, 0, 0, -75243, -7945702]\) | \(-15777367606441/3574920\) | \(-10674653921280\) | \([]\) | \(69120\) | \(1.4921\) |
Rank
sage: E.rank()
The elliptic curves in class 22320bf have rank \(1\).
Complex multiplication
The elliptic curves in class 22320bf do not have complex multiplication.Modular form 22320.2.a.bf
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.