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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 20631a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
20631.g4 | 20631a1 | \([1, 1, 0, 254, 3415]\) | \(12167/39\) | \(-5773399671\) | \([2]\) | \(12672\) | \(0.55623\) | \(\Gamma_0(N)\)-optimal |
20631.g3 | 20631a2 | \([1, 1, 0, -2391, 37800]\) | \(10218313/1521\) | \(225162587169\) | \([2, 2]\) | \(25344\) | \(0.90281\) | |
20631.g2 | 20631a3 | \([1, 1, 0, -10326, -370059]\) | \(822656953/85683\) | \(12684159077187\) | \([2]\) | \(50688\) | \(1.2494\) | |
20631.g1 | 20631a4 | \([1, 1, 0, -36776, 2699199]\) | \(37159393753/1053\) | \(155881791117\) | \([2]\) | \(50688\) | \(1.2494\) |
Rank
sage: E.rank()
The elliptic curves in class 20631a have rank \(1\).
Complex multiplication
The elliptic curves in class 20631a do not have complex multiplication.Modular form 20631.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.