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SageMath
E = EllipticCurve("gf1")
E.isogeny_class()
Elliptic curves in class 388080gf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388080.gf4 | 388080gf1 | \([0, 0, 0, 1809717, -1247568518]\) | \(1865864036231/2993760000\) | \(-1051701995130716160000\) | \([2]\) | \(11796480\) | \(2.7189\) | \(\Gamma_0(N)\)-optimal |
| 388080.gf3 | 388080gf2 | \([0, 0, 0, -12302283, -12726269318]\) | \(586145095611769/140040608400\) | \(49195990077227075174400\) | \([2, 2]\) | \(23592960\) | \(3.0655\) | |
| 388080.gf2 | 388080gf3 | \([0, 0, 0, -66633483, 198654697402]\) | \(93137706732176569/5369647977540\) | \(1886346764980637377904640\) | \([2]\) | \(47185920\) | \(3.4120\) | |
| 388080.gf1 | 388080gf4 | \([0, 0, 0, -183763083, -958744087238]\) | \(1953542217204454969/170843779260\) | \(60017083367876405084160\) | \([2]\) | \(47185920\) | \(3.4120\) |
Rank
sage: E.rank()
The elliptic curves in class 388080gf have rank \(1\).
Complex multiplication
The elliptic curves in class 388080gf do not have complex multiplication.Modular form 388080.2.a.gf
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.
