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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 6675.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6675.h1 | 6675e3 | \([1, 1, 0, -59375, -5593500]\) | \(1481582988342001/2919645\) | \(45619453125\) | \([2]\) | \(21504\) | \(1.2978\) | |
| 6675.h2 | 6675e2 | \([1, 1, 0, -3750, -86625]\) | \(373403541601/16040025\) | \(250625390625\) | \([2, 2]\) | \(10752\) | \(0.95121\) | |
| 6675.h3 | 6675e1 | \([1, 1, 0, -625, 4000]\) | \(1732323601/500625\) | \(7822265625\) | \([2]\) | \(5376\) | \(0.60464\) | \(\Gamma_0(N)\)-optimal |
| 6675.h4 | 6675e4 | \([1, 1, 0, 1875, -317250]\) | \(46617130799/2823400845\) | \(-44115638203125\) | \([4]\) | \(21504\) | \(1.2978\) |
Rank
sage: E.rank()
The elliptic curves in class 6675.h have rank \(0\).
Complex multiplication
The elliptic curves in class 6675.h do not have complex multiplication.Modular form 6675.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
