Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 20097.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20097.k1 | 20097f3 | \([1, -1, 0, -78137136, -265829047781]\) | \(72371679832051361738355457/1627857\) | \(1186707753\) | \([2]\) | \(663552\) | \(2.6864\) | |
| 20097.k2 | 20097f4 | \([1, -1, 0, -4897926, -4127010503]\) | \(17825137625614555960417/216318148151991039\) | \(157695930002801467431\) | \([2]\) | \(663552\) | \(2.6864\) | |
| 20097.k3 | 20097f2 | \([1, -1, 0, -4883571, -4152662888]\) | \(17668869054438249282097/2649918412449\) | \(1931790522675321\) | \([2, 2]\) | \(331776\) | \(2.3398\) | |
| 20097.k4 | 20097f1 | \([1, -1, 0, -304326, -65228801]\) | \(-4275768267198290017/52843101620463\) | \(-38522621081317527\) | \([2]\) | \(165888\) | \(1.9932\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 20097.k have rank \(0\).
Complex multiplication
The elliptic curves in class 20097.k do not have complex multiplication.Modular form 20097.2.a.k
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.
