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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 105966bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
105966.a2 | 105966bf1 | \([1, -1, 0, 3366, 177214]\) | \(237176659/907578\) | \(-16136356564818\) | \([]\) | \(403200\) | \(1.2171\) | \(\Gamma_0(N)\)-optimal |
105966.a1 | 105966bf2 | \([1, -1, 0, -334629, -75305291]\) | \(-233073901963421/3214155168\) | \(-57146332156024608\) | \([]\) | \(2016000\) | \(2.0218\) |
Rank
sage: E.rank()
The elliptic curves in class 105966bf have rank \(2\).
Complex multiplication
The elliptic curves in class 105966bf do not have complex multiplication.Modular form 105966.2.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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.