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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 55770.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
55770.p1 | 55770q2 | \([1, 1, 0, -4956097, -4248825719]\) | \(2789222297765780449/677605500\) | \(3270672325849500\) | \([2]\) | \(1548288\) | \(2.3542\) | |
55770.p2 | 55770q1 | \([1, 1, 0, -308597, -67005219]\) | \(-673350049820449/10617750000\) | \(-51249851259750000\) | \([2]\) | \(774144\) | \(2.0077\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 55770.p have rank \(1\).
Complex multiplication
The elliptic curves in class 55770.p do not have complex multiplication.Modular form 55770.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.