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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 429b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 429.b5 | 429b1 | \([1, 0, 0, -24, 63]\) | \(-1532808577/938223\) | \(-938223\) | \([4]\) | \(64\) | \(-0.15205\) | \(\Gamma_0(N)\)-optimal |
| 429.b4 | 429b2 | \([1, 0, 0, -429, 3384]\) | \(8732907467857/1656369\) | \(1656369\) | \([2, 4]\) | \(128\) | \(0.19452\) | |
| 429.b3 | 429b3 | \([1, 0, 0, -474, 2619]\) | \(11779205551777/3763454409\) | \(3763454409\) | \([2, 2]\) | \(256\) | \(0.54110\) | |
| 429.b1 | 429b4 | \([1, 0, 0, -6864, 218313]\) | \(35765103905346817/1287\) | \(1287\) | \([4]\) | \(256\) | \(0.54110\) | |
| 429.b2 | 429b5 | \([1, 0, 0, -3009, -61770]\) | \(3013001140430737/108679952667\) | \(108679952667\) | \([2]\) | \(512\) | \(0.88767\) | |
| 429.b6 | 429b6 | \([1, 0, 0, 1341, 18228]\) | \(266679605718863/296110251723\) | \(-296110251723\) | \([2]\) | \(512\) | \(0.88767\) |
Rank
sage: E.rank()
The elliptic curves in class 429b have rank \(1\).
Complex multiplication
The elliptic curves in class 429b do not have complex multiplication.Modular form 429.2.a.b
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
