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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 55545.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
55545.p1 | 55545r1 | \([0, 1, 1, -11285, -1431244]\) | \(-1073741824/5325075\) | \(-788302211616675\) | \([]\) | \(228096\) | \(1.5428\) | \(\Gamma_0(N)\)-optimal |
55545.p2 | 55545r2 | \([0, 1, 1, 99805, 34972949]\) | \(742692847616/3992296875\) | \(-591003217042546875\) | \([]\) | \(684288\) | \(2.0921\) |
Rank
sage: E.rank()
The elliptic curves in class 55545.p have rank \(0\).
Complex multiplication
The elliptic curves in class 55545.p do not have complex multiplication.Modular form 55545.2.a.p
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.