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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 4650.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4650.h1 | 4650a5 | \([1, 1, 0, -7688000, -8208007500]\) | \(3216206300355197383681/57660\) | \(900937500\) | \([2]\) | \(98304\) | \(2.1865\) | |
| 4650.h2 | 4650a3 | \([1, 1, 0, -480500, -128400000]\) | \(785209010066844481/3324675600\) | \(51948056250000\) | \([2, 2]\) | \(49152\) | \(1.8400\) | |
| 4650.h3 | 4650a6 | \([1, 1, 0, -473000, -132592500]\) | \(-749011598724977281/51173462246460\) | \(-799585347600937500\) | \([2]\) | \(98304\) | \(2.1865\) | |
| 4650.h4 | 4650a4 | \([1, 1, 0, -92500, 8404000]\) | \(5601911201812801/1271193750000\) | \(19862402343750000\) | \([2]\) | \(49152\) | \(1.8400\) | |
| 4650.h5 | 4650a2 | \([1, 1, 0, -30500, -1950000]\) | \(200828550012481/12454560000\) | \(194602500000000\) | \([2, 2]\) | \(24576\) | \(1.4934\) | |
| 4650.h6 | 4650a1 | \([1, 1, 0, 1500, -126000]\) | \(23862997439/457113600\) | \(-7142400000000\) | \([2]\) | \(12288\) | \(1.1468\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4650.h have rank \(1\).
Complex multiplication
The elliptic curves in class 4650.h do not have complex multiplication.Modular form 4650.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
