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SageMath
E = EllipticCurve("do1")
E.isogeny_class()
Elliptic curves in class 485184do
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 485184.do4 | 485184do1 | \([0, -1, 0, -6294877, 2383163053]\) | \(572616640141312/280535480757\) | \(13514791776226930603008\) | \([2]\) | \(35389440\) | \(2.9398\) | \(\Gamma_0(N)\)-optimal |
| 485184.do2 | 485184do2 | \([0, -1, 0, -53665297, -149647462895]\) | \(22174957026242512/278654127129\) | \(214786521580663695753216\) | \([2, 2]\) | \(70778880\) | \(3.2864\) | |
| 485184.do3 | 485184do3 | \([0, -1, 0, -9218977, -390057607775]\) | \(-28104147578308/21301741002339\) | \(-65677503515122815183028224\) | \([2]\) | \(141557760\) | \(3.6329\) | |
| 485184.do1 | 485184do4 | \([0, -1, 0, -856038337, -9639955305407]\) | \(22501000029889239268/3620708343\) | \(11163363825450036953088\) | \([2]\) | \(141557760\) | \(3.6329\) |
Rank
sage: E.rank()
The elliptic curves in class 485184do have rank \(0\).
Complex multiplication
The elliptic curves in class 485184do do not have complex multiplication.Modular form 485184.2.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.
