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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 128a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
128.a2 | 128a1 | \([0, 1, 0, 1, 1]\) | \(128\) | \(-256\) | \([2]\) | \(4\) | \(-0.85643\) | \(\Gamma_0(N)\)-optimal |
128.a1 | 128a2 | \([0, 1, 0, -9, 7]\) | \(10976\) | \(8192\) | \([2]\) | \(8\) | \(-0.50986\) |
Rank
sage: E.rank()
The elliptic curves in class 128a have rank \(1\).
Complex multiplication
The elliptic curves in class 128a do not have complex multiplication.Modular form 128.2.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 Cremona 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 Cremona labels.