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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 10710.m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10710.m1 | 10710m3 | \([1, -1, 0, -100854, -12302672]\) | \(155624507032726369/175394100\) | \(127862298900\) | \([2]\) | \(32768\) | \(1.4165\) | |
| 10710.m2 | 10710m4 | \([1, -1, 0, -15534, 486040]\) | \(568671957006049/191329687500\) | \(139479342187500\) | \([4]\) | \(32768\) | \(1.4165\) | |
| 10710.m3 | 10710m2 | \([1, -1, 0, -6354, -187772]\) | \(38920307374369/1274490000\) | \(929103210000\) | \([2, 2]\) | \(16384\) | \(1.0699\) | |
| 10710.m4 | 10710m1 | \([1, -1, 0, 126, -10220]\) | \(302111711/61689600\) | \(-44971718400\) | \([2]\) | \(8192\) | \(0.72334\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 10710.m have rank \(1\).
Complex multiplication
The elliptic curves in class 10710.m do not have complex multiplication.Modular form 10710.2.a.m
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
