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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 73689r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 73689.g3 | 73689r1 | \([1, 1, 1, -5992, -178936]\) | \(13430356633/180873\) | \(320427552753\) | \([2]\) | \(99840\) | \(1.0144\) | \(\Gamma_0(N)\)-optimal |
| 73689.g2 | 73689r2 | \([1, 1, 1, -11437, 189146]\) | \(93391282153/44876601\) | \(79501636144161\) | \([2, 2]\) | \(199680\) | \(1.3610\) | |
| 73689.g4 | 73689r3 | \([1, 1, 1, 41198, 1494494]\) | \(4365111505607/3058314567\) | \(-5417990812629087\) | \([2]\) | \(399360\) | \(1.7076\) | |
| 73689.g1 | 73689r4 | \([1, 1, 1, -151192, 22549946]\) | \(215751695207833/163381911\) | \(289441021633071\) | \([2]\) | \(399360\) | \(1.7076\) |
Rank
sage: E.rank()
The elliptic curves in class 73689r have rank \(0\).
Complex multiplication
The elliptic curves in class 73689r do not have complex multiplication.Modular form 73689.2.a.r
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.
