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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 36822.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
36822.r1 | 36822p3 | \([1, 1, 1, -270938, 40925927]\) | \(46753267515625/11591221248\) | \(545319215478079488\) | \([2]\) | \(497664\) | \(2.1136\) | |
36822.r2 | 36822p1 | \([1, 1, 1, -92243, -10817647]\) | \(1845026709625/793152\) | \(37314534606912\) | \([2]\) | \(165888\) | \(1.5643\) | \(\Gamma_0(N)\)-optimal |
36822.r3 | 36822p2 | \([1, 1, 1, -77803, -14300575]\) | \(-1107111813625/1228691592\) | \(-57804878422932552\) | \([2]\) | \(331776\) | \(1.9109\) | |
36822.r4 | 36822p4 | \([1, 1, 1, 653222, 260506343]\) | \(655215969476375/1001033261568\) | \(-47094491700770001408\) | \([2]\) | \(995328\) | \(2.4602\) |
Rank
sage: E.rank()
The elliptic curves in class 36822.r have rank \(1\).
Complex multiplication
The elliptic curves in class 36822.r do not have complex multiplication.Modular form 36822.2.a.r
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.