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SageMath
E = EllipticCurve("eh1")
E.isogeny_class()
Elliptic curves in class 54450eh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 54450.ea3 | 54450eh1 | \([1, -1, 1, -4742255, 843152247]\) | \(15781142246787/8722841600\) | \(6519253776076800000000\) | \([2]\) | \(4976640\) | \(2.8763\) | \(\Gamma_0(N)\)-optimal |
| 54450.ea4 | 54450eh2 | \([1, -1, 1, 18489745, 6651152247]\) | \(935355271080573/566899520000\) | \(-423687830857335000000000\) | \([2]\) | \(9953280\) | \(3.2228\) | |
| 54450.ea1 | 54450eh3 | \([1, -1, 1, -292238255, 1922956224247]\) | \(5066026756449723/11000000\) | \(5993218543640625000000\) | \([2]\) | \(14929920\) | \(3.4256\) | |
| 54450.ea2 | 54450eh4 | \([1, -1, 1, -288971255, 1968047358247]\) | \(-4898016158612283/236328125000\) | \(-128760554648529052734375000\) | \([2]\) | \(29859840\) | \(3.7721\) |
Rank
sage: E.rank()
The elliptic curves in class 54450eh have rank \(1\).
Complex multiplication
The elliptic curves in class 54450eh do not have complex multiplication.Modular form 54450.2.a.eh
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 Cremona 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 Cremona labels.
