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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 208725p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 208725.y2 | 208725p1 | \([1, 0, 0, -69638, 102852267]\) | \(-1349232625/164333367\) | \(-4548852874623234375\) | \([2]\) | \(2764800\) | \(2.2596\) | \(\Gamma_0(N)\)-optimal |
| 208725.y1 | 208725p2 | \([1, 0, 0, -3745013, 2767499142]\) | \(209849322390625/1882056627\) | \(52096533127886671875\) | \([2]\) | \(5529600\) | \(2.6062\) |
Rank
sage: E.rank()
The elliptic curves in class 208725p have rank \(1\).
Complex multiplication
The elliptic curves in class 208725p do not have complex multiplication.Modular form 208725.2.a.p
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.
