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SageMath
E = EllipticCurve("du1")
E.isogeny_class()
Elliptic curves in class 189630du
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
189630.ba1 | 189630du1 | \([1, -1, 0, -84240, 8697856]\) | \(770842973809/66873600\) | \(5735489269305600\) | \([2]\) | \(1474560\) | \(1.7646\) | \(\Gamma_0(N)\)-optimal |
189630.ba2 | 189630du2 | \([1, -1, 0, 92160, 40202896]\) | \(1009328859791/8734528080\) | \(-749126592187177680\) | \([2]\) | \(2949120\) | \(2.1111\) |
Rank
sage: E.rank()
The elliptic curves in class 189630du have rank \(1\).
Complex multiplication
The elliptic curves in class 189630du do not have complex multiplication.Modular form 189630.2.a.du
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.