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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 108900.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
108900.e1 | 108900bi2 | \([0, 0, 0, -39830175, -96751388250]\) | \(1016339184/25\) | \(171894386673900000000\) | \([2]\) | \(9732096\) | \(2.9922\) | |
108900.e2 | 108900bi1 | \([0, 0, 0, -2395800, -1630641375]\) | \(-3538944/625\) | \(-268584979177968750000\) | \([2]\) | \(4866048\) | \(2.6456\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 108900.e have rank \(1\).
Complex multiplication
The elliptic curves in class 108900.e do not have complex multiplication.Modular form 108900.2.a.e
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.