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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 68445bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68445.x2 | 68445bf1 | \([1, -1, 0, -285, -57394]\) | \(-9/5\) | \(-1425091223205\) | \([]\) | \(82944\) | \(1.0112\) | \(\Gamma_0(N)\)-optimal |
68445.x1 | 68445bf2 | \([1, -1, 0, -342510, 77572925]\) | \(-15590912409/78125\) | \(-22267050362578125\) | \([]\) | \(580608\) | \(1.9841\) |
Rank
sage: E.rank()
The elliptic curves in class 68445bf have rank \(0\).
Complex multiplication
The elliptic curves in class 68445bf do not have complex multiplication.Modular form 68445.2.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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.