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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 25047e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25047.b2 | 25047e1 | \([1, -1, 1, -25070, -22211076]\) | \(-1349232625/164333367\) | \(-212231279718421623\) | \([2]\) | \(153600\) | \(2.0042\) | \(\Gamma_0(N)\)-optimal |
| 25047.b1 | 25047e2 | \([1, -1, 1, -1348205, -597510174]\) | \(209849322390625/1882056627\) | \(2430615849614680563\) | \([2]\) | \(307200\) | \(2.3507\) |
Rank
sage: E.rank()
The elliptic curves in class 25047e have rank \(1\).
Complex multiplication
The elliptic curves in class 25047e do not have complex multiplication.Modular form 25047.2.a.e
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.
