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SageMath
E = EllipticCurve("zr1")
E.isogeny_class()
Elliptic curves in class 235200zr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
235200.zr2 | 235200zr1 | \([0, 1, 0, 126992, 30251738]\) | \(1925134784/4465125\) | \(-525317491125000000\) | \([2]\) | \(2654208\) | \(2.0843\) | \(\Gamma_0(N)\)-optimal |
235200.zr1 | 235200zr2 | \([0, 1, 0, -1030633, 332391863]\) | \(16079333824/2953125\) | \(22235661000000000000\) | \([2]\) | \(5308416\) | \(2.4308\) |
Rank
sage: E.rank()
The elliptic curves in class 235200zr have rank \(0\).
Complex multiplication
The elliptic curves in class 235200zr do not have complex multiplication.Modular form 235200.2.a.zr
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.