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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 41382.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 41382.p1 | 41382m3 | \([1, -1, 0, -93132, -87972912]\) | \(-69173457625/2550136832\) | \(-3293420035095134208\) | \([]\) | \(583200\) | \(2.2334\) | |
| 41382.p2 | 41382m1 | \([1, -1, 0, -16902, 850284]\) | \(-413493625/152\) | \(-196303131288\) | \([]\) | \(64800\) | \(1.1348\) | \(\Gamma_0(N)\)-optimal |
| 41382.p3 | 41382m2 | \([1, -1, 0, 10323, 3212325]\) | \(94196375/3511808\) | \(-4535387545277952\) | \([]\) | \(194400\) | \(1.6841\) |
Rank
sage: E.rank()
The elliptic curves in class 41382.p have rank \(1\).
Complex multiplication
The elliptic curves in class 41382.p do not have complex multiplication.Modular form 41382.2.a.p
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.
