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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 16170p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 16170.p4 | 16170p1 | \([1, 1, 0, 12568, -716736]\) | \(1865864036231/2993760000\) | \(-352212870240000\) | \([2]\) | \(61440\) | \(1.4764\) | \(\Gamma_0(N)\)-optimal |
| 16170.p3 | 16170p2 | \([1, 1, 0, -85432, -7400336]\) | \(586145095611769/140040608400\) | \(16475637537651600\) | \([2, 2]\) | \(122880\) | \(1.8230\) | |
| 16170.p1 | 16170p3 | \([1, 1, 0, -1276132, -555360476]\) | \(1953542217204454969/170843779260\) | \(20099599786159740\) | \([2]\) | \(245760\) | \(2.1696\) | |
| 16170.p2 | 16170p4 | \([1, 1, 0, -462732, 114769404]\) | \(93137706732176569/5369647977540\) | \(631733714909603460\) | \([2]\) | \(245760\) | \(2.1696\) |
Rank
sage: E.rank()
The elliptic curves in class 16170p have rank \(0\).
Complex multiplication
The elliptic curves in class 16170p do not have complex multiplication.Modular form 16170.2.a.p
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.
