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SageMath
E = EllipticCurve("he1")
E.isogeny_class()
Elliptic curves in class 106722.he
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
106722.he1 | 106722gy2 | \([1, -1, 1, -57127874, 166207834433]\) | \(46546832455691959/748268928\) | \(331463315979518760576\) | \([2]\) | \(10321920\) | \(3.0704\) | |
106722.he2 | 106722gy1 | \([1, -1, 1, -3461954, 2762908481]\) | \(-10358806345399/1445216256\) | \(-640192522495421693952\) | \([2]\) | \(5160960\) | \(2.7238\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 106722.he have rank \(1\).
Complex multiplication
The elliptic curves in class 106722.he do not have complex multiplication.Modular form 106722.2.a.he
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.