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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 5202.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5202.j1 | 5202h3 | \([1, -1, 1, -1952105, 794166441]\) | \(46753267515625/11591221248\) | \(203962465044874395648\) | \([2]\) | \(165888\) | \(2.6073\) | |
| 5202.j2 | 5202h1 | \([1, -1, 1, -664610, -208300575]\) | \(1845026709625/793152\) | \(13956530861938752\) | \([2]\) | \(55296\) | \(2.0580\) | \(\Gamma_0(N)\)-optimal |
| 5202.j3 | 5202h2 | \([1, -1, 1, -560570, -275801727]\) | \(-1107111813625/1228691592\) | \(-21620410871500869192\) | \([2]\) | \(110592\) | \(2.4046\) | |
| 5202.j4 | 5202h4 | \([1, -1, 1, 4706455, 5031674025]\) | \(655215969476375/1001033261568\) | \(-17614469368924240531968\) | \([2]\) | \(331776\) | \(2.9539\) |
Rank
sage: E.rank()
The elliptic curves in class 5202.j have rank \(1\).
Complex multiplication
The elliptic curves in class 5202.j do not have complex multiplication.Modular form 5202.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
