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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 127050dd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 127050.do3 | 127050dd1 | \([1, 0, 1, -600216026, 5645768759948]\) | \(863913648706111516969/2486234429521920\) | \(68820561753098158080000000\) | \([2]\) | \(72253440\) | \(3.8289\) | \(\Gamma_0(N)\)-optimal |
| 127050.do2 | 127050dd2 | \([1, 0, 1, -848024026, 539932727948]\) | \(2436531580079063806249/1405478914998681600\) | \(38904556752093427718400000000\) | \([2, 2]\) | \(144506880\) | \(4.1754\) | |
| 127050.do4 | 127050dd3 | \([1, 0, 1, 3386007974, 4316689271948]\) | \(155099895405729262880471/90047655797243760000\) | \(-2492576799247202386083750000000\) | \([2]\) | \(289013760\) | \(4.5220\) | |
| 127050.do1 | 127050dd4 | \([1, 0, 1, -9046984026, -330009338632052]\) | \(2958414657792917260183849/12401051653985258880\) | \(343269054206027800104870000000\) | \([2]\) | \(289013760\) | \(4.5220\) |
Rank
sage: E.rank()
The elliptic curves in class 127050dd have rank \(1\).
Complex multiplication
The elliptic curves in class 127050dd do not have complex multiplication.Modular form 127050.2.a.dd
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.
