Show commands:
SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 129360.cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 129360.cp1 | 129360bk4 | \([0, -1, 0, -4312800, -3445926000]\) | \(73639964854838596/9904125\) | \(1193175451776000\) | \([2]\) | \(2949120\) | \(2.3063\) | |
| 129360.cp2 | 129360bk3 | \([0, -1, 0, -496680, 49407072]\) | \(112477694831716/56396484375\) | \(6794229750000000000\) | \([4]\) | \(2949120\) | \(2.3063\) | |
| 129360.cp3 | 129360bk2 | \([0, -1, 0, -270300, -53460000]\) | \(72516235474384/833765625\) | \(25111473156000000\) | \([2, 2]\) | \(1474560\) | \(1.9597\) | |
| 129360.cp4 | 129360bk1 | \([0, -1, 0, -3495, -2126718]\) | \(-2508888064/1037680875\) | \(-1953313876206000\) | \([2]\) | \(737280\) | \(1.6132\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 129360.cp have rank \(1\).
Complex multiplication
The elliptic curves in class 129360.cp do not have complex multiplication.Modular form 129360.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
