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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 169065.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 169065.j1 | 169065j4 | \([1, -1, 1, -67913562713, -6811772649438058]\) | \(1968666709544018637994033129/113621848881699526875\) | \(1999322753404114876898034151875\) | \([2]\) | \(552075264\) | \(4.8775\) | |
| 169065.j2 | 169065j3 | \([1, -1, 1, -22347293963, 1202514916766942]\) | \(70141892778055497175333129/5090453819946781723125\) | \(89573090453483405703834197098125\) | \([2]\) | \(552075264\) | \(4.8775\) | |
| 169065.j3 | 169065j2 | \([1, -1, 1, -4489803338, -93446035866808]\) | \(568832774079017834683129/114800389711906640625\) | \(2020060697037668724880578515625\) | \([2, 2]\) | \(276037632\) | \(4.5309\) | |
| 169065.j4 | 169065j1 | \([1, -1, 1, 590274787, -8722524929308]\) | \(1292603583867446566871/2615843353271484375\) | \(-46029132486497953948974609375\) | \([2]\) | \(138018816\) | \(4.1843\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 169065.j have rank \(1\).
Complex multiplication
The elliptic curves in class 169065.j do not have complex multiplication.Modular form 169065.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
