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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 34650f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
34650.bx1 | 34650f1 | \([1, -1, 0, -5817, 159341]\) | \(51603494067/4336640\) | \(1829520000000\) | \([2]\) | \(61440\) | \(1.0947\) | \(\Gamma_0(N)\)-optimal |
34650.bx2 | 34650f2 | \([1, -1, 0, 6183, 723341]\) | \(61958108493/573927200\) | \(-242125537500000\) | \([2]\) | \(122880\) | \(1.4413\) |
Rank
sage: E.rank()
The elliptic curves in class 34650f have rank \(1\).
Complex multiplication
The elliptic curves in class 34650f do not have complex multiplication.Modular form 34650.2.a.f
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.