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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 5070q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5070.s3 | 5070q1 | \([1, 1, 1, -2285, 39827]\) | \(273359449/9360\) | \(45178932240\) | \([4]\) | \(5376\) | \(0.81636\) | \(\Gamma_0(N)\)-optimal |
| 5070.s2 | 5070q2 | \([1, 1, 1, -5665, -110245]\) | \(4165509529/1368900\) | \(6607418840100\) | \([2, 2]\) | \(10752\) | \(1.1629\) | |
| 5070.s1 | 5070q3 | \([1, 1, 1, -81715, -9023305]\) | \(12501706118329/2570490\) | \(12407264266410\) | \([2]\) | \(21504\) | \(1.5095\) | |
| 5070.s4 | 5070q4 | \([1, 1, 1, 16305, -734193]\) | \(99317171591/106616250\) | \(-514616275046250\) | \([2]\) | \(21504\) | \(1.5095\) |
Rank
sage: E.rank()
The elliptic curves in class 5070q have rank \(1\).
Complex multiplication
The elliptic curves in class 5070q do not have complex multiplication.Modular form 5070.2.a.q
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.
