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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1610.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1610.e1 | 1610b1 | \([1, -1, 1, -1348, 6631]\) | \(270701905514769/139540889600\) | \(139540889600\) | \([2]\) | \(1536\) | \(0.83058\) | \(\Gamma_0(N)\)-optimal |
1610.e2 | 1610b2 | \([1, -1, 1, 5052, 47591]\) | \(14262456319278831/9284810958080\) | \(-9284810958080\) | \([2]\) | \(3072\) | \(1.1772\) |
Rank
sage: E.rank()
The elliptic curves in class 1610.e have rank \(1\).
Complex multiplication
The elliptic curves in class 1610.e do not have complex multiplication.Modular form 1610.2.a.e
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.