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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1610.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1610.d1 | 1610g2 | \([1, 0, 0, -25120, -1116288]\) | \(1753007192038126081/478174101507200\) | \(478174101507200\) | \([2]\) | \(8960\) | \(1.5245\) | |
1610.d2 | 1610g1 | \([1, 0, 0, -9120, 320512]\) | \(83890194895342081/3958384640000\) | \(3958384640000\) | \([2]\) | \(4480\) | \(1.1780\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1610.d have rank \(1\).
Complex multiplication
The elliptic curves in class 1610.d do not have complex multiplication.Modular form 1610.2.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.