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SageMath
E = EllipticCurve("eq1")
E.isogeny_class()
Elliptic curves in class 68400.eq
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68400.eq1 | 68400ei2 | \([0, 0, 0, -83973675, -296184851750]\) | \(1403607530712116449/39475350\) | \(1841761929600000000\) | \([2]\) | \(5160960\) | \(3.0152\) | |
68400.eq2 | 68400ei1 | \([0, 0, 0, -5241675, -4640255750]\) | \(-341370886042369/1817528220\) | \(-84798596632320000000\) | \([2]\) | \(2580480\) | \(2.6687\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 68400.eq have rank \(1\).
Complex multiplication
The elliptic curves in class 68400.eq do not have complex multiplication.Modular form 68400.2.a.eq
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.