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SageMath
E = EllipticCurve("iq1")
E.isogeny_class()
Elliptic curves in class 277200iq
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
277200.iq2 | 277200iq1 | \([0, 0, 0, -166200, -26079125]\) | \(75216478666752/326095\) | \(2201141250000\) | \([2]\) | \(995328\) | \(1.5752\) | \(\Gamma_0(N)\)-optimal |
277200.iq3 | 277200iq2 | \([0, 0, 0, -163575, -26942750]\) | \(-4481782160112/310023175\) | \(-33482502900000000\) | \([2]\) | \(1990656\) | \(1.9217\) | |
277200.iq1 | 277200iq3 | \([0, 0, 0, -232200, -3479625]\) | \(281370820608/161767375\) | \(796016810531250000\) | \([2]\) | \(2985984\) | \(2.1245\) | |
277200.iq4 | 277200iq4 | \([0, 0, 0, 925425, -27789750]\) | \(1113258734352/648484375\) | \(-51056471812500000000\) | \([2]\) | \(5971968\) | \(2.4711\) |
Rank
sage: E.rank()
The elliptic curves in class 277200iq have rank \(1\).
Complex multiplication
The elliptic curves in class 277200iq do not have complex multiplication.Modular form 277200.2.a.iq
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.