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SageMath
E = EllipticCurve("fg1")
E.isogeny_class()
Elliptic curves in class 76230fg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 76230.fc3 | 76230fg1 | \([1, -1, 1, -1079222, -297159739]\) | \(107639597521009/32699842560\) | \(42230799257582960640\) | \([2]\) | \(2457600\) | \(2.4708\) | \(\Gamma_0(N)\)-optimal |
| 76230.fc2 | 76230fg2 | \([1, -1, 1, -6654902, 6380274629]\) | \(25238585142450289/995844326400\) | \(1286101049655981081600\) | \([2, 2]\) | \(4915200\) | \(2.8174\) | |
| 76230.fc4 | 76230fg3 | \([1, -1, 1, 2928298, 23239040069]\) | \(2150235484224911/181905111732960\) | \(-234924625200483921558240\) | \([2]\) | \(9830400\) | \(3.1640\) | |
| 76230.fc1 | 76230fg4 | \([1, -1, 1, -105448982, 416810400581]\) | \(100407751863770656369/166028940000\) | \(214421057937022860000\) | \([2]\) | \(9830400\) | \(3.1640\) |
Rank
sage: E.rank()
The elliptic curves in class 76230fg have rank \(0\).
Complex multiplication
The elliptic curves in class 76230fg do not have complex multiplication.Modular form 76230.2.a.fg
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.
