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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 3264h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3264.m5 | 3264h1 | \([0, -1, 0, -2177, 36993]\) | \(4354703137/352512\) | \(92408905728\) | \([2]\) | \(3072\) | \(0.84708\) | \(\Gamma_0(N)\)-optimal |
| 3264.m4 | 3264h2 | \([0, -1, 0, -7297, -195455]\) | \(163936758817/30338064\) | \(7952941449216\) | \([2, 2]\) | \(6144\) | \(1.1937\) | |
| 3264.m2 | 3264h3 | \([0, -1, 0, -110977, -14192255]\) | \(576615941610337/27060804\) | \(7093827403776\) | \([2, 2]\) | \(12288\) | \(1.5402\) | |
| 3264.m6 | 3264h4 | \([0, -1, 0, 14463, -1157247]\) | \(1276229915423/2927177028\) | \(-767341894828032\) | \([2]\) | \(12288\) | \(1.5402\) | |
| 3264.m1 | 3264h5 | \([0, -1, 0, -1775617, -910101503]\) | \(2361739090258884097/5202\) | \(1363673088\) | \([2]\) | \(24576\) | \(1.8868\) | |
| 3264.m3 | 3264h6 | \([0, -1, 0, -105217, -15737087]\) | \(-491411892194497/125563633938\) | \(-32915753255043072\) | \([4]\) | \(24576\) | \(1.8868\) |
Rank
sage: E.rank()
The elliptic curves in class 3264h have rank \(0\).
Complex multiplication
The elliptic curves in class 3264h do not have complex multiplication.Modular form 3264.2.a.h
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.
