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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 7650p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7650.j2 | 7650p1 | \([1, -1, 0, -57492, -5289584]\) | \(1845026709625/793152\) | \(9034497000000\) | \([2]\) | \(27648\) | \(1.4461\) | \(\Gamma_0(N)\)-optimal |
| 7650.j3 | 7650p2 | \([1, -1, 0, -48492, -7008584]\) | \(-1107111813625/1228691592\) | \(-13995565165125000\) | \([2]\) | \(55296\) | \(1.7927\) | |
| 7650.j1 | 7650p3 | \([1, -1, 0, -168867, 20235541]\) | \(46753267515625/11591221248\) | \(132031254528000000\) | \([2]\) | \(82944\) | \(1.9955\) | |
| 7650.j4 | 7650p4 | \([1, -1, 0, 407133, 127947541]\) | \(655215969476375/1001033261568\) | \(-11402394495048000000\) | \([2]\) | \(165888\) | \(2.3420\) |
Rank
sage: E.rank()
The elliptic curves in class 7650p have rank \(0\).
Complex multiplication
The elliptic curves in class 7650p do not have complex multiplication.Modular form 7650.2.a.p
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.
