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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 336.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 336.d1 | 336d3 | \([0, 1, 0, -21504, -1220940]\) | \(268498407453697/252\) | \(1032192\) | \([2]\) | \(384\) | \(0.88225\) | |
| 336.d2 | 336d5 | \([0, 1, 0, -14624, 669300]\) | \(84448510979617/933897762\) | \(3825245233152\) | \([8]\) | \(768\) | \(1.2288\) | |
| 336.d3 | 336d4 | \([0, 1, 0, -1664, -9804]\) | \(124475734657/63011844\) | \(258096513024\) | \([2, 4]\) | \(384\) | \(0.88225\) | |
| 336.d4 | 336d2 | \([0, 1, 0, -1344, -19404]\) | \(65597103937/63504\) | \(260112384\) | \([2, 2]\) | \(192\) | \(0.53568\) | |
| 336.d5 | 336d1 | \([0, 1, 0, -64, -460]\) | \(-7189057/16128\) | \(-66060288\) | \([2]\) | \(96\) | \(0.18910\) | \(\Gamma_0(N)\)-optimal |
| 336.d6 | 336d6 | \([0, 1, 0, 6176, -69388]\) | \(6359387729183/4218578658\) | \(-17279298183168\) | \([4]\) | \(768\) | \(1.2288\) |
Rank
sage: E.rank()
The elliptic curves in class 336.d have rank \(0\).
Complex multiplication
The elliptic curves in class 336.d do not have complex multiplication.Modular form 336.2.a.d
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
