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SageMath
E = EllipticCurve("dj1")
E.isogeny_class()
Elliptic curves in class 203280dj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
203280.bp3 | 203280dj1 | \([0, -1, 0, -322021, 71484976]\) | \(-130287139815424/2250652635\) | \(-63794694923411760\) | \([2]\) | \(2488320\) | \(2.0225\) | \(\Gamma_0(N)\)-optimal |
203280.bp2 | 203280dj2 | \([0, -1, 0, -5173516, 4530979180]\) | \(33766427105425744/9823275\) | \(4455047905862400\) | \([2]\) | \(4976640\) | \(2.3691\) | |
203280.bp4 | 203280dj3 | \([0, -1, 0, 1246139, 341639740]\) | \(7549996227362816/6152409907875\) | \(-174389911180879086000\) | \([2]\) | \(7464960\) | \(2.5718\) | |
203280.bp1 | 203280dj4 | \([0, -1, 0, -6001156, 2985452956]\) | \(52702650535889104/22020583921875\) | \(9986766764344524000000\) | \([2]\) | \(14929920\) | \(2.9184\) |
Rank
sage: E.rank()
The elliptic curves in class 203280dj have rank \(0\).
Complex multiplication
The elliptic curves in class 203280dj do not have complex multiplication.Modular form 203280.2.a.dj
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.