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SageMath
E = EllipticCurve("er1")
E.isogeny_class()
Elliptic curves in class 134640er
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 134640.b4 | 134640er1 | \([0, 0, 0, 661902, 486271703]\) | \(2749522520524826624/10348942908211875\) | \(-120710070081383310000\) | \([2]\) | \(4521984\) | \(2.5361\) | \(\Gamma_0(N)\)-optimal |
| 134640.b3 | 134640er2 | \([0, 0, 0, -6719223, 5870064278]\) | \(179768412409071136336/24269556653741025\) | \(4529281740947765049600\) | \([2, 2]\) | \(9043968\) | \(2.8827\) | |
| 134640.b1 | 134640er3 | \([0, 0, 0, -103749123, 406739393138]\) | \(165443431757918996551684/3760316927358255\) | \(2807061545005227924480\) | \([2]\) | \(18087936\) | \(3.2293\) | |
| 134640.b2 | 134640er4 | \([0, 0, 0, -27787323, -50436539782]\) | \(3178603484863787328484/367337349353309445\) | \(274215861942848087454720\) | \([2]\) | \(18087936\) | \(3.2293\) |
Rank
sage: E.rank()
The elliptic curves in class 134640er have rank \(1\).
Complex multiplication
The elliptic curves in class 134640er do not have complex multiplication.Modular form 134640.2.a.er
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.
