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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 82810p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 82810.c1 | 82810p1 | \([1, 0, 1, -6964494, 7072736192]\) | \(65787589563409/10400000\) | \(5905840221226400000\) | \([2]\) | \(3870720\) | \(2.6124\) | \(\Gamma_0(N)\)-optimal |
| 82810.c2 | 82810p2 | \([1, 0, 1, -6302014, 8472423936]\) | \(-48743122863889/26406250000\) | \(-14995297436707656250000\) | \([2]\) | \(7741440\) | \(2.9590\) |
Rank
sage: E.rank()
The elliptic curves in class 82810p have rank \(0\).
Complex multiplication
The elliptic curves in class 82810p do not have complex multiplication.Modular form 82810.2.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.
