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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 380880h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 380880.h8 | 380880h1 | \([0, 0, 0, 112677, -44701558]\) | \(357911/2160\) | \(-954790839316316160\) | \([2]\) | \(4866048\) | \(2.1319\) | \(\Gamma_0(N)\)-optimal |
| 380880.h6 | 380880h2 | \([0, 0, 0, -1410843, -584332342]\) | \(702595369/72900\) | \(32224190826925670400\) | \([2, 2]\) | \(9732096\) | \(2.4785\) | |
| 380880.h7 | 380880h3 | \([0, 0, 0, -1029963, 1316639738]\) | \(-273359449/1536000\) | \(-678962374624935936000\) | \([2]\) | \(14598144\) | \(2.6812\) | |
| 380880.h5 | 380880h4 | \([0, 0, 0, -5219643, 3953471978]\) | \(35578826569/5314410\) | \(2349143511282881372160\) | \([2]\) | \(19464192\) | \(2.8251\) | |
| 380880.h4 | 380880h5 | \([0, 0, 0, -21978363, -39658506838]\) | \(2656166199049/33750\) | \(14918606864317440000\) | \([2]\) | \(19464192\) | \(2.8251\) | |
| 380880.h3 | 380880h6 | \([0, 0, 0, -25406283, 49196607482]\) | \(4102915888729/9000000\) | \(3978295163817984000000\) | \([2, 2]\) | \(29196288\) | \(3.0278\) | |
| 380880.h1 | 380880h7 | \([0, 0, 0, -406286283, 3152073615482]\) | \(16778985534208729/81000\) | \(35804656474361856000\) | \([2]\) | \(58392576\) | \(3.3744\) | |
| 380880.h2 | 380880h8 | \([0, 0, 0, -34547403, 10637535098]\) | \(10316097499609/5859375000\) | \(2590035913944000000000000\) | \([2]\) | \(58392576\) | \(3.3744\) |
Rank
sage: E.rank()
The elliptic curves in class 380880h have rank \(0\).
Complex multiplication
The elliptic curves in class 380880h do not have complex multiplication.Modular form 380880.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
