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SageMath
E = EllipticCurve("v1")
E.isogeny_class()
Elliptic curves in class 8112.v
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8112.v1 | 8112bg3 | \([0, 1, 0, -56079664, -161661424300]\) | \(986551739719628473/111045168\) | \(2195430671601303552\) | \([2]\) | \(645120\) | \(2.9437\) | |
| 8112.v2 | 8112bg4 | \([0, 1, 0, -6326064, 2074990932]\) | \(1416134368422073/725251155408\) | \(14338657501936570662912\) | \([4]\) | \(645120\) | \(2.9437\) | |
| 8112.v3 | 8112bg2 | \([0, 1, 0, -3513904, -2513329324]\) | \(242702053576633/2554695936\) | \(50507896160863125504\) | \([2, 2]\) | \(322560\) | \(2.5971\) | |
| 8112.v4 | 8112bg1 | \([0, 1, 0, -52784, -97467564]\) | \(-822656953/207028224\) | \(-4093074206135156736\) | \([2]\) | \(161280\) | \(2.2506\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8112.v have rank \(1\).
Complex multiplication
The elliptic curves in class 8112.v do not have complex multiplication.Modular form 8112.2.a.v
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.
