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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 3234.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3234.e1 | 3234g2 | \([1, 1, 0, -72265, 7447189]\) | \(121681065322255375/12702096\) | \(4356818928\) | \([2]\) | \(8192\) | \(1.2785\) | |
3234.e2 | 3234g1 | \([1, 1, 0, -4505, 115557]\) | \(-29489309167375/303595776\) | \(-104133351168\) | \([2]\) | \(4096\) | \(0.93194\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3234.e have rank \(1\).
Complex multiplication
The elliptic curves in class 3234.e do not have complex multiplication.Modular form 3234.2.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.