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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 10710e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10710.k3 | 10710e1 | \([1, -1, 0, -1239, 23373]\) | \(-7794190562283/4000066000\) | \(-108001782000\) | \([6]\) | \(11520\) | \(0.82174\) | \(\Gamma_0(N)\)-optimal |
| 10710.k2 | 10710e2 | \([1, -1, 0, -21819, 1245825]\) | \(42547659109328043/6195437500\) | \(167276812500\) | \([6]\) | \(23040\) | \(1.1683\) | |
| 10710.k4 | 10710e3 | \([1, -1, 0, 9786, -295372]\) | \(5265299629773/4930293760\) | \(-97042972078080\) | \([2]\) | \(34560\) | \(1.3710\) | |
| 10710.k1 | 10710e4 | \([1, -1, 0, -50694, -2629900]\) | \(731992986690867/270340772800\) | \(5321117431022400\) | \([2]\) | \(69120\) | \(1.7176\) |
Rank
sage: E.rank()
The elliptic curves in class 10710e have rank \(1\).
Complex multiplication
The elliptic curves in class 10710e do not have complex multiplication.Modular form 10710.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
