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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 18050.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
18050.j1 | 18050e3 | \([1, 0, 1, -771826, 2099324548]\) | \(-69173457625/2550136832\) | \(-1874584905187328000000\) | \([]\) | \(933120\) | \(2.7621\) | |
18050.j2 | 18050e1 | \([1, 0, 1, -140076, -20196702]\) | \(-413493625/152\) | \(-111733967375000\) | \([]\) | \(103680\) | \(1.6635\) | \(\Gamma_0(N)\)-optimal |
18050.j3 | 18050e2 | \([1, 0, 1, 85549, -76693202]\) | \(94196375/3511808\) | \(-2581501582232000000\) | \([]\) | \(311040\) | \(2.2128\) |
Rank
sage: E.rank()
The elliptic curves in class 18050.j have rank \(0\).
Complex multiplication
The elliptic curves in class 18050.j do not have complex multiplication.Modular form 18050.2.a.j
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.