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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 10710k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10710.o4 | 10710k1 | \([1, -1, 0, 140616, -6858432]\) | \(421792317902132351/271682182840320\) | \(-198056311290593280\) | \([2]\) | \(143360\) | \(2.0079\) | \(\Gamma_0(N)\)-optimal |
| 10710.o3 | 10710k2 | \([1, -1, 0, -596664, -55961280]\) | \(32224493437735955329/16782725759385600\) | \(12234607078592102400\) | \([2, 2]\) | \(286720\) | \(2.3544\) | |
| 10710.o1 | 10710k3 | \([1, -1, 0, -7595064, -8046734400]\) | \(66464620505913166201729/74880071980801920\) | \(54587572474004599680\) | \([2]\) | \(573440\) | \(2.7010\) | |
| 10710.o2 | 10710k4 | \([1, -1, 0, -5394744, 4783382208]\) | \(23818189767728437646209/232359312482640000\) | \(169389938799844560000\) | \([2]\) | \(573440\) | \(2.7010\) |
Rank
sage: E.rank()
The elliptic curves in class 10710k have rank \(0\).
Complex multiplication
The elliptic curves in class 10710k do not have complex multiplication.Modular form 10710.2.a.k
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.
