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SageMath
E = EllipticCurve("mg1")
E.isogeny_class()
Elliptic curves in class 187200mg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
187200.hs1 | 187200mg1 | \([0, 0, 0, -586200, -162295000]\) | \(1909913257984/129730653\) | \(1513178336592000000\) | \([2]\) | \(2457600\) | \(2.2370\) | \(\Gamma_0(N)\)-optimal |
187200.hs2 | 187200mg2 | \([0, 0, 0, 507300, -698110000]\) | \(77366117936/1172914587\) | \(-218894011884288000000\) | \([2]\) | \(4915200\) | \(2.5836\) |
Rank
sage: E.rank()
The elliptic curves in class 187200mg have rank \(0\).
Complex multiplication
The elliptic curves in class 187200mg do not have complex multiplication.Modular form 187200.2.a.mg
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.