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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 98736cf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
98736.bi2 | 98736cf1 | \([0, -1, 0, -494688, 134035200]\) | \(1845026709625/793152\) | \(5755359847514112\) | \([2]\) | \(829440\) | \(1.9842\) | \(\Gamma_0(N)\)-optimal |
98736.bi3 | 98736cf2 | \([0, -1, 0, -417248, 177339648]\) | \(-1107111813625/1228691592\) | \(-8915771823780298752\) | \([2]\) | \(1658880\) | \(2.3308\) | |
98736.bi1 | 98736cf3 | \([0, -1, 0, -1453008, -509189184]\) | \(46753267515625/11591221248\) | \(84109539349824012288\) | \([2]\) | \(2488320\) | \(2.5335\) | |
98736.bi4 | 98736cf4 | \([0, -1, 0, 3503152, -3233094720]\) | \(655215969476375/1001033261568\) | \(-7263811526232750686208\) | \([2]\) | \(4976640\) | \(2.8801\) |
Rank
sage: E.rank()
The elliptic curves in class 98736cf have rank \(0\).
Complex multiplication
The elliptic curves in class 98736cf do not have complex multiplication.Modular form 98736.2.a.cf
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.