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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 10710o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10710.j4 | 10710o1 | \([1, -1, 0, -99, 4293]\) | \(-148035889/10710000\) | \(-7807590000\) | \([2]\) | \(8192\) | \(0.57791\) | \(\Gamma_0(N)\)-optimal |
| 10710.j3 | 10710o2 | \([1, -1, 0, -4599, 120393]\) | \(14758408587889/114704100\) | \(83619288900\) | \([2, 2]\) | \(16384\) | \(0.92449\) | |
| 10710.j2 | 10710o3 | \([1, -1, 0, -7749, -62937]\) | \(70593496254289/38358689670\) | \(27963484769430\) | \([2]\) | \(32768\) | \(1.2711\) | |
| 10710.j1 | 10710o4 | \([1, -1, 0, -73449, 7680123]\) | \(60111445514713489/3673530\) | \(2678003370\) | \([2]\) | \(32768\) | \(1.2711\) |
Rank
sage: E.rank()
The elliptic curves in class 10710o have rank \(1\).
Complex multiplication
The elliptic curves in class 10710o do not have complex multiplication.Modular form 10710.2.a.o
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
