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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 390390.cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 390390.cm1 | 390390cm3 | \([1, 0, 1, -16364443, 25478629958]\) | \(100407751863770656369/166028940000\) | \(801389981852460000\) | \([2]\) | \(19660800\) | \(2.6982\) | |
| 390390.cm2 | 390390cm2 | \([1, 0, 1, -1032763, 389868806]\) | \(25238585142450289/995844326400\) | \(4806750357266457600\) | \([2, 2]\) | \(9830400\) | \(2.3516\) | |
| 390390.cm3 | 390390cm1 | \([1, 0, 1, -167483, -18197242]\) | \(107639597521009/32699842560\) | \(157835894367191040\) | \([2]\) | \(4915200\) | \(2.0050\) | \(\Gamma_0(N)\)-optimal |
| 390390.cm4 | 390390cm4 | \([1, 0, 1, 454437, 1420795846]\) | \(2150235484224911/181905111732960\) | \(-878021230458656924640\) | \([2]\) | \(19660800\) | \(2.6982\) |
Rank
sage: E.rank()
The elliptic curves in class 390390.cm have rank \(1\).
Complex multiplication
The elliptic curves in class 390390.cm do not have complex multiplication.Modular form 390390.2.a.cm
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
