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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 88725h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 88725.h6 | 88725h1 | \([1, 1, 1, 4137, -20844]\) | \(103823/63\) | \(-4751390109375\) | \([2]\) | \(147456\) | \(1.1217\) | \(\Gamma_0(N)\)-optimal |
| 88725.h5 | 88725h2 | \([1, 1, 1, -16988, -189844]\) | \(7189057/3969\) | \(299337576890625\) | \([2, 2]\) | \(294912\) | \(1.4683\) | |
| 88725.h3 | 88725h3 | \([1, 1, 1, -164863, 25540406]\) | \(6570725617/45927\) | \(3463763389734375\) | \([2]\) | \(589824\) | \(1.8148\) | |
| 88725.h2 | 88725h4 | \([1, 1, 1, -207113, -36313594]\) | \(13027640977/21609\) | \(1629726807515625\) | \([2, 2]\) | \(589824\) | \(1.8148\) | |
| 88725.h4 | 88725h5 | \([1, 1, 1, -143738, -58875094]\) | \(-4354703137/17294403\) | \(-1304324688281671875\) | \([2]\) | \(1179648\) | \(2.1614\) | |
| 88725.h1 | 88725h6 | \([1, 1, 1, -3312488, -2321869594]\) | \(53297461115137/147\) | \(11086576921875\) | \([2]\) | \(1179648\) | \(2.1614\) |
Rank
sage: E.rank()
The elliptic curves in class 88725h have rank \(1\).
Complex multiplication
The elliptic curves in class 88725h do not have complex multiplication.Modular form 88725.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
