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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 770.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
770.a1 | 770b4 | \([1, 0, 1, -15649, 580116]\) | \(423783056881319689/99207416000000\) | \(99207416000000\) | \([2]\) | \(3456\) | \(1.3975\) | |
770.a2 | 770b2 | \([1, 0, 1, -14634, 680132]\) | \(346553430870203929/8300600\) | \(8300600\) | \([6]\) | \(1152\) | \(0.84821\) | |
770.a3 | 770b1 | \([1, 0, 1, -914, 10596]\) | \(-84309998289049/414124480\) | \(-414124480\) | \([6]\) | \(576\) | \(0.50164\) | \(\Gamma_0(N)\)-optimal |
770.a4 | 770b3 | \([1, 0, 1, 2271, 56852]\) | \(1296134247276791/2137096192000\) | \(-2137096192000\) | \([2]\) | \(1728\) | \(1.0509\) |
Rank
sage: E.rank()
The elliptic curves in class 770.a have rank \(0\).
Complex multiplication
The elliptic curves in class 770.a do not have complex multiplication.Modular form 770.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.