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SageMath
E = EllipticCurve("hr1")
E.isogeny_class()
Elliptic curves in class 378560.hr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 378560.hr1 | 378560hr4 | \([0, -1, 0, -158654721, -769112665855]\) | \(349046010201856969/7245875000\) | \(9168343443144704000000\) | \([2]\) | \(55738368\) | \(3.3329\) | |
| 378560.hr2 | 378560hr3 | \([0, -1, 0, -10259201, -11138028799]\) | \(94376601570889/12235496000\) | \(15481805789531734016000\) | \([2]\) | \(27869184\) | \(2.9863\) | |
| 378560.hr3 | 378560hr2 | \([0, -1, 0, -3282881, 542764481]\) | \(3092354182009/1689383150\) | \(2137608629221680742400\) | \([2]\) | \(18579456\) | \(2.7836\) | |
| 378560.hr4 | 378560hr1 | \([0, -1, 0, -2525761, 1543828545]\) | \(1408317602329/2153060\) | \(2724307765962997760\) | \([2]\) | \(9289728\) | \(2.4370\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 378560.hr have rank \(1\).
Complex multiplication
The elliptic curves in class 378560.hr do not have complex multiplication.Modular form 378560.2.a.hr
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
