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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 5610.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5610.h1 | 5610i3 | \([1, 1, 0, -376907, 86267439]\) | \(5921450764096952391481/200074809015963750\) | \(200074809015963750\) | \([2]\) | \(73728\) | \(2.0921\) | |
| 5610.h2 | 5610i2 | \([1, 1, 0, -58157, -3556311]\) | \(21754112339458491481/7199734626562500\) | \(7199734626562500\) | \([2, 2]\) | \(36864\) | \(1.7456\) | |
| 5610.h3 | 5610i1 | \([1, 1, 0, -52377, -4634859]\) | \(15891267085572193561/3334993530000\) | \(3334993530000\) | \([2]\) | \(18432\) | \(1.3990\) | \(\Gamma_0(N)\)-optimal |
| 5610.h4 | 5610i4 | \([1, 1, 0, 168113, -24237389]\) | \(525440531549759128199/559322204589843750\) | \(-559322204589843750\) | \([2]\) | \(73728\) | \(2.0921\) |
Rank
sage: E.rank()
The elliptic curves in class 5610.h have rank \(1\).
Complex multiplication
The elliptic curves in class 5610.h do not have complex multiplication.Modular form 5610.2.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.
