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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 80223.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
80223.d1 | 80223d1 | \([1, 1, 1, -13373, 584090]\) | \(149298747625/1611753\) | \(2855318756433\) | \([2]\) | \(184320\) | \(1.2054\) | \(\Gamma_0(N)\)-optimal |
80223.d2 | 80223d2 | \([1, 1, 1, -3088, 1472714]\) | \(-1838265625/528749793\) | \(-936712512036873\) | \([2]\) | \(368640\) | \(1.5520\) |
Rank
sage: E.rank()
The elliptic curves in class 80223.d have rank \(0\).
Complex multiplication
The elliptic curves in class 80223.d do not have complex multiplication.Modular form 80223.2.a.d
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.