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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 12480s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12480.bq4 | 12480s1 | \([0, -1, 0, -18465, 1601697]\) | \(-2656166199049/2658140160\) | \(-696815494103040\) | \([2]\) | \(61440\) | \(1.5438\) | \(\Gamma_0(N)\)-optimal |
| 12480.bq3 | 12480s2 | \([0, -1, 0, -346145, 78475425]\) | \(17496824387403529/6580454400\) | \(1725026638233600\) | \([2, 2]\) | \(122880\) | \(1.8904\) | |
| 12480.bq2 | 12480s3 | \([0, -1, 0, -397345, 53786785]\) | \(26465989780414729/10571870144160\) | \(2771352327070679040\) | \([2]\) | \(245760\) | \(2.2370\) | |
| 12480.bq1 | 12480s4 | \([0, -1, 0, -5537825, 5017839777]\) | \(71647584155243142409/10140000\) | \(2658140160000\) | \([4]\) | \(245760\) | \(2.2370\) |
Rank
sage: E.rank()
The elliptic curves in class 12480s have rank \(1\).
Complex multiplication
The elliptic curves in class 12480s do not have complex multiplication.Modular form 12480.2.a.s
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.
