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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 67760p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 67760.bm4 | 67760p1 | \([0, 0, 0, -5687, 407286]\) | \(-44851536/132055\) | \(-59889532890880\) | \([2]\) | \(122880\) | \(1.3305\) | \(\Gamma_0(N)\)-optimal |
| 67760.bm3 | 67760p2 | \([0, 0, 0, -124267, 16842474]\) | \(116986321764/148225\) | \(268891780326400\) | \([2, 2]\) | \(245760\) | \(1.6771\) | |
| 67760.bm2 | 67760p3 | \([0, 0, 0, -158147, 6929186]\) | \(120564797922/64054375\) | \(232399324424960000\) | \([2]\) | \(491520\) | \(2.0237\) | |
| 67760.bm1 | 67760p4 | \([0, 0, 0, -1987667, 1078607794]\) | \(239369344910082/385\) | \(1396840417280\) | \([2]\) | \(491520\) | \(2.0237\) |
Rank
sage: E.rank()
The elliptic curves in class 67760p have rank \(0\).
Complex multiplication
The elliptic curves in class 67760p do not have complex multiplication.Modular form 67760.2.a.p
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.
