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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 47775cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47775.df6 | 47775cp1 | \([1, 0, 1, -134776, -19054927]\) | \(147281603041/5265\) | \(9678468515625\) | \([2]\) | \(221184\) | \(1.5808\) | \(\Gamma_0(N)\)-optimal |
| 47775.df5 | 47775cp2 | \([1, 0, 1, -140901, -17229677]\) | \(168288035761/27720225\) | \(50957136734765625\) | \([2, 2]\) | \(442368\) | \(1.9274\) | |
| 47775.df7 | 47775cp3 | \([1, 0, 1, 257224, -96854677]\) | \(1023887723039/2798036865\) | \(-5143534986412265625\) | \([2]\) | \(884736\) | \(2.2739\) | |
| 47775.df4 | 47775cp4 | \([1, 0, 1, -637026, 179235823]\) | \(15551989015681/1445900625\) | \(2657949416103515625\) | \([2, 2]\) | \(884736\) | \(2.2739\) | |
| 47775.df8 | 47775cp5 | \([1, 0, 1, 741099, 849004573]\) | \(24487529386319/183539412225\) | \(-337394192325922265625\) | \([2]\) | \(1769472\) | \(2.6205\) | |
| 47775.df2 | 47775cp6 | \([1, 0, 1, -9953151, 12085243573]\) | \(59319456301170001/594140625\) | \(1092188287353515625\) | \([2, 2]\) | \(1769472\) | \(2.6205\) | |
| 47775.df3 | 47775cp7 | \([1, 0, 1, -9714276, 12692941573]\) | \(-55150149867714721/5950927734375\) | \(-10939385890960693359375\) | \([2]\) | \(3538944\) | \(2.9671\) | |
| 47775.df1 | 47775cp8 | \([1, 0, 1, -159250026, 773499306073]\) | \(242970740812818720001/24375\) | \(44807724609375\) | \([2]\) | \(3538944\) | \(2.9671\) |
Rank
sage: E.rank()
The elliptic curves in class 47775cp have rank \(0\).
Complex multiplication
The elliptic curves in class 47775cp do not have complex multiplication.Modular form 47775.2.a.cp
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
