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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 7728.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7728.h1 | 7728a2 | \([0, -1, 0, -792, 6480]\) | \(26860713266/7394247\) | \(15143417856\) | \([2]\) | \(5120\) | \(0.66095\) | |
| 7728.h2 | 7728a1 | \([0, -1, 0, 128, 592]\) | \(224727548/299943\) | \(-307141632\) | \([2]\) | \(2560\) | \(0.31437\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 7728.h have rank \(1\).
Complex multiplication
The elliptic curves in class 7728.h do not have complex multiplication.Modular form 7728.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
