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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 34.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
34.a1 | 34a4 | \([1, 0, 0, -113, -329]\) | \(159661140625/48275138\) | \(48275138\) | \([2]\) | \(12\) | \(0.17949\) | |
34.a2 | 34a3 | \([1, 0, 0, -103, -411]\) | \(120920208625/19652\) | \(19652\) | \([2]\) | \(6\) | \(-0.16709\) | |
34.a3 | 34a2 | \([1, 0, 0, -43, 105]\) | \(8805624625/2312\) | \(2312\) | \([6]\) | \(4\) | \(-0.36982\) | |
34.a4 | 34a1 | \([1, 0, 0, -3, 1]\) | \(3048625/1088\) | \(1088\) | \([6]\) | \(2\) | \(-0.71639\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 34.a have rank \(0\).
Complex multiplication
The elliptic curves in class 34.a do not have complex multiplication.Modular form 34.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.