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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 4002a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4002.b2 | 4002a1 | \([1, 1, 0, 20, 124]\) | \(817400375/6267132\) | \(-6267132\) | \([2]\) | \(896\) | \(-0.014264\) | \(\Gamma_0(N)\)-optimal |
4002.b1 | 4002a2 | \([1, 1, 0, -270, 1458]\) | \(2189403771625/201304602\) | \(201304602\) | \([2]\) | \(1792\) | \(0.33231\) |
Rank
sage: E.rank()
The elliptic curves in class 4002a have rank \(1\).
Complex multiplication
The elliptic curves in class 4002a do not have complex multiplication.Modular form 4002.2.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.