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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 2100.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2100.a1 | 2100c4 | \([0, -1, 0, -45708, -3746088]\) | \(2640279346000/3087\) | \(12348000000\) | \([2]\) | \(5184\) | \(1.2195\) | |
2100.a2 | 2100c3 | \([0, -1, 0, -2833, -58838]\) | \(-10061824000/352947\) | \(-88236750000\) | \([2]\) | \(2592\) | \(0.87295\) | |
2100.a3 | 2100c2 | \([0, -1, 0, -708, -2088]\) | \(9826000/5103\) | \(20412000000\) | \([2]\) | \(1728\) | \(0.67022\) | |
2100.a4 | 2100c1 | \([0, -1, 0, 167, -338]\) | \(2048000/1323\) | \(-330750000\) | \([2]\) | \(864\) | \(0.32365\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2100.a have rank \(0\).
Complex multiplication
The elliptic curves in class 2100.a do not have complex multiplication.Modular form 2100.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.