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SageMath
E = EllipticCurve("ej1")
E.isogeny_class()
Elliptic curves in class 87120.ej
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
87120.ej1 | 87120fu1 | \([0, 0, 0, -17787, -2494294]\) | \(-117649/440\) | \(-2327535232450560\) | \([]\) | \(345600\) | \(1.6343\) | \(\Gamma_0(N)\)-optimal |
87120.ej2 | 87120fu2 | \([0, 0, 0, 156453, 59221514]\) | \(80062991/332750\) | \(-1760198519540736000\) | \([]\) | \(1036800\) | \(2.1836\) |
Rank
sage: E.rank()
The elliptic curves in class 87120.ej have rank \(1\).
Complex multiplication
The elliptic curves in class 87120.ej do not have complex multiplication.Modular form 87120.2.a.ej
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.