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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1980.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1980.a1 | 1980b4 | \([0, 0, 0, -63903, 6217702]\) | \(154639330142416/33275\) | \(6209913600\) | \([6]\) | \(5184\) | \(1.2638\) | |
1980.a2 | 1980b3 | \([0, 0, 0, -4008, 96433]\) | \(610462990336/8857805\) | \(103317437520\) | \([6]\) | \(2592\) | \(0.91718\) | |
1980.a3 | 1980b2 | \([0, 0, 0, -903, 5902]\) | \(436334416/171875\) | \(32076000000\) | \([2]\) | \(1728\) | \(0.71445\) | |
1980.a4 | 1980b1 | \([0, 0, 0, -408, -3107]\) | \(643956736/15125\) | \(176418000\) | \([2]\) | \(864\) | \(0.36788\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1980.a have rank \(0\).
Complex multiplication
The elliptic curves in class 1980.a do not have complex multiplication.Modular form 1980.2.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.