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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 32760j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 32760.k5 | 32760j1 | \([0, 0, 0, 582, -92783]\) | \(1869154304/319921875\) | \(-3731568750000\) | \([2]\) | \(49152\) | \(1.0916\) | \(\Gamma_0(N)\)-optimal |
| 32760.k4 | 32760j2 | \([0, 0, 0, -27543, -1707158]\) | \(12381975627856/419225625\) | \(78237563040000\) | \([2, 2]\) | \(98304\) | \(1.4382\) | |
| 32760.k3 | 32760j3 | \([0, 0, 0, -68043, 4473142]\) | \(46670944188964/15429366225\) | \(11517960169497600\) | \([2, 2]\) | \(196608\) | \(1.7847\) | |
| 32760.k2 | 32760j4 | \([0, 0, 0, -437043, -111207458]\) | \(12367124507424964/14926275\) | \(11142404582400\) | \([2]\) | \(196608\) | \(1.7847\) | |
| 32760.k6 | 32760j5 | \([0, 0, 0, 196557, 30774382]\) | \(562511980386718/599562079935\) | \(-895141388846315520\) | \([2]\) | \(393216\) | \(2.1313\) | |
| 32760.k1 | 32760j6 | \([0, 0, 0, -980643, 373711102]\) | \(69855246474511682/14613770535\) | \(21818242498590720\) | \([2]\) | \(393216\) | \(2.1313\) |
Rank
sage: E.rank()
The elliptic curves in class 32760j have rank \(1\).
Complex multiplication
The elliptic curves in class 32760j do not have complex multiplication.Modular form 32760.2.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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
