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SageMath
E = EllipticCurve("do1")
E.isogeny_class()
Elliptic curves in class 154560do
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 154560.j4 | 154560do1 | \([0, -1, 0, 5999, -197519]\) | \(1457028215984/1851148215\) | \(-30329212354560\) | \([2]\) | \(393216\) | \(1.2729\) | \(\Gamma_0(N)\)-optimal |
| 154560.j3 | 154560do2 | \([0, -1, 0, -36321, -1881855]\) | \(80859142234084/23148101025\) | \(1517033948774400\) | \([2, 2]\) | \(786432\) | \(1.6195\) | |
| 154560.j2 | 154560do3 | \([0, -1, 0, -216641, 37391841]\) | \(8579021289461282/374333754375\) | \(49064673853440000\) | \([2]\) | \(1572864\) | \(1.9661\) | |
| 154560.j1 | 154560do4 | \([0, -1, 0, -533121, -149630175]\) | \(127847420666360642/17899707105\) | \(2346150409666560\) | \([2]\) | \(1572864\) | \(1.9661\) |
Rank
sage: E.rank()
The elliptic curves in class 154560do have rank \(0\).
Complex multiplication
The elliptic curves in class 154560do do not have complex multiplication.Modular form 154560.2.a.do
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.
