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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 100018.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
100018.a1 | 100018a1 | \([1, -1, 0, -1031, -12483]\) | \(121265324086473/51209216\) | \(51209216\) | \([2]\) | \(52040\) | \(0.44030\) | \(\Gamma_0(N)\)-optimal |
100018.a2 | 100018a2 | \([1, -1, 0, -871, -16611]\) | \(-73123777485513/80028802592\) | \(-80028802592\) | \([2]\) | \(104080\) | \(0.78688\) |
Rank
sage: E.rank()
The elliptic curves in class 100018.a have rank \(0\).
Complex multiplication
The elliptic curves in class 100018.a do not have complex multiplication.Modular form 100018.2.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.