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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 62.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
62.a1 | 62a4 | \([1, -1, 1, -331, 2397]\) | \(3999236143617/62\) | \(62\) | \([2]\) | \(8\) | \(-0.10808\) | |
62.a2 | 62a3 | \([1, -1, 1, -31, 5]\) | \(3196010817/1847042\) | \(1847042\) | \([2]\) | \(8\) | \(-0.10808\) | |
62.a3 | 62a2 | \([1, -1, 1, -21, 41]\) | \(979146657/3844\) | \(3844\) | \([2, 2]\) | \(4\) | \(-0.45465\) | |
62.a4 | 62a1 | \([1, -1, 1, -1, 1]\) | \(-35937/496\) | \(-496\) | \([4]\) | \(2\) | \(-0.80122\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 62.a have rank \(0\).
Complex multiplication
The elliptic curves in class 62.a do not have complex multiplication.Modular form 62.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.