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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 129591p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 129591.d3 | 129591p1 | \([1, -1, 1, -145710401, 677027876352]\) | \(264918160154242157473/536027170833\) | \(692261921644510548177\) | \([4]\) | \(11059200\) | \(3.2498\) | \(\Gamma_0(N)\)-optimal |
| 129591.d2 | 129591p2 | \([1, -1, 1, -147284006, 661658161596]\) | \(273594167224805799793/11903648120953281\) | \(15373180262458200156956289\) | \([2, 2]\) | \(22118400\) | \(3.5964\) | |
| 129591.d4 | 129591p3 | \([1, -1, 1, 74964559, 2483296299762]\) | \(36075142039228937567/2083708275110728497\) | \(-2691042494045745782131012593\) | \([2]\) | \(44236800\) | \(3.9430\) | |
| 129591.d1 | 129591p4 | \([1, -1, 1, -394710251, -2143660604214]\) | \(5265932508006615127873/1510137598013239041\) | \(1950294336616696259391777729\) | \([2]\) | \(44236800\) | \(3.9430\) |
Rank
sage: E.rank()
The elliptic curves in class 129591p have rank \(0\).
Complex multiplication
The elliptic curves in class 129591p do not have complex multiplication.Modular form 129591.2.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.
