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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 41405.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 41405.h1 | 41405d3 | \([0, -1, 1, -1087571, 457033527]\) | \(-250523582464/13671875\) | \(-7763837430248046875\) | \([]\) | \(590976\) | \(2.3829\) | |
| 41405.h2 | 41405d1 | \([0, -1, 1, -11041, -491723]\) | \(-262144/35\) | \(-19875423821435\) | \([]\) | \(65664\) | \(1.2843\) | \(\Gamma_0(N)\)-optimal |
| 41405.h3 | 41405d2 | \([0, -1, 1, 71769, 1239006]\) | \(71991296/42875\) | \(-24347394181257875\) | \([]\) | \(196992\) | \(1.8336\) |
Rank
sage: E.rank()
The elliptic curves in class 41405.h have rank \(0\).
Complex multiplication
The elliptic curves in class 41405.h do not have complex multiplication.Modular form 41405.2.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.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
