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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 1584j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1584.k4 | 1584j1 | \([0, 0, 0, -1035, -12358]\) | \(1108717875/45056\) | \(4982833152\) | \([2]\) | \(768\) | \(0.62663\) | \(\Gamma_0(N)\)-optimal |
| 1584.k2 | 1584j2 | \([0, 0, 0, -16395, -808006]\) | \(4406910829875/7744\) | \(856424448\) | \([2]\) | \(1536\) | \(0.97321\) | |
| 1584.k3 | 1584j3 | \([0, 0, 0, -12555, 537786]\) | \(2714704875/21296\) | \(1716916912128\) | \([2]\) | \(2304\) | \(1.1759\) | |
| 1584.k1 | 1584j4 | \([0, 0, 0, -21195, -296838]\) | \(13060888875/7086244\) | \(571304102510592\) | \([2]\) | \(4608\) | \(1.5225\) |
Rank
sage: E.rank()
The elliptic curves in class 1584j have rank \(1\).
Complex multiplication
The elliptic curves in class 1584j do not have complex multiplication.Modular form 1584.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
