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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1002.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1002.a1 | 1002a2 | \([1, 1, 0, -860, -10074]\) | \(70470585447625/4518018\) | \(4518018\) | \([2]\) | \(384\) | \(0.33467\) | |
1002.a2 | 1002a1 | \([1, 1, 0, -50, -192]\) | \(-14260515625/4382748\) | \(-4382748\) | \([2]\) | \(192\) | \(-0.011904\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1002.a have rank \(0\).
Complex multiplication
The elliptic curves in class 1002.a do not have complex multiplication.Modular form 1002.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.