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SageMath
E = EllipticCurve("gb1")
E.isogeny_class()
Elliptic curves in class 58800gb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 58800.m6 | 58800gb1 | \([0, -1, 0, 19192, -231888]\) | \(103823/63\) | \(-474360768000000\) | \([2]\) | \(196608\) | \(1.5053\) | \(\Gamma_0(N)\)-optimal |
| 58800.m5 | 58800gb2 | \([0, -1, 0, -78808, -1799888]\) | \(7189057/3969\) | \(29884728384000000\) | \([2, 2]\) | \(393216\) | \(1.8519\) | |
| 58800.m3 | 58800gb3 | \([0, -1, 0, -764808, 256136112]\) | \(6570725617/45927\) | \(345808999872000000\) | \([2]\) | \(786432\) | \(2.1985\) | |
| 58800.m2 | 58800gb4 | \([0, -1, 0, -960808, -361655888]\) | \(13027640977/21609\) | \(162705743424000000\) | \([2, 2]\) | \(786432\) | \(2.1985\) | |
| 58800.m4 | 58800gb5 | \([0, -1, 0, -666808, -587447888]\) | \(-4354703137/17294403\) | \(-130218829987008000000\) | \([2]\) | \(1572864\) | \(2.5450\) | |
| 58800.m1 | 58800gb6 | \([0, -1, 0, -15366808, -23180759888]\) | \(53297461115137/147\) | \(1106841792000000\) | \([2]\) | \(1572864\) | \(2.5450\) |
Rank
sage: E.rank()
The elliptic curves in class 58800gb have rank \(1\).
Complex multiplication
The elliptic curves in class 58800gb do not have complex multiplication.Modular form 58800.2.a.gb
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
