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SageMath
E = EllipticCurve("de1")
E.isogeny_class()
Elliptic curves in class 88200.de
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
88200.de1 | 88200et2 | \([0, 0, 0, -1356075, -604280250]\) | \(3721734/25\) | \(1852548213600000000\) | \([2]\) | \(1658880\) | \(2.3400\) | |
88200.de2 | 88200et1 | \([0, 0, 0, -33075, -20837250]\) | \(-108/5\) | \(-185254821360000000\) | \([2]\) | \(829440\) | \(1.9934\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 88200.de have rank \(1\).
Complex multiplication
The elliptic curves in class 88200.de do not have complex multiplication.Modular form 88200.2.a.de
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.