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SageMath
E = EllipticCurve("fv1")
E.isogeny_class()
Elliptic curves in class 232050fv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
232050.fv4 | 232050fv1 | \([1, 0, 0, -135688, -131008]\) | \(17681870665400761/10232167895040\) | \(159877623360000000\) | \([2]\) | \(2654208\) | \(1.9906\) | \(\Gamma_0(N)\)-optimal |
232050.fv2 | 232050fv2 | \([1, 0, 0, -1487688, 696148992]\) | \(23304472877725373881/82743765249600\) | \(1292871332025000000\) | \([2, 2]\) | \(5308416\) | \(2.3371\) | |
232050.fv1 | 232050fv3 | \([1, 0, 0, -23782688, 44639593992]\) | \(95210863233510962017081/1206641250360\) | \(18853769536875000\) | \([2]\) | \(10616832\) | \(2.6837\) | |
232050.fv3 | 232050fv4 | \([1, 0, 0, -824688, 1320031992]\) | \(-3969837635175430201/45883867071315000\) | \(-716935422989296875000\) | \([2]\) | \(10616832\) | \(2.6837\) |
Rank
sage: E.rank()
The elliptic curves in class 232050fv have rank \(1\).
Complex multiplication
The elliptic curves in class 232050fv do not have complex multiplication.Modular form 232050.2.a.fv
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.