Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 270802.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
270802.e1 | 270802e2 | \([1, -1, 0, -336033061, -2272124875275]\) | \(7054751972146948898193/332947845138448288\) | \(198045142965045515454924448\) | \([2]\) | \(90316800\) | \(3.8066\) | |
270802.e2 | 270802e1 | \([1, -1, 0, -332130821, -2329675891243]\) | \(6811821555839776164753/16312107262976\) | \(9702821814671604663296\) | \([2]\) | \(45158400\) | \(3.4600\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 270802.e have rank \(0\).
Complex multiplication
The elliptic curves in class 270802.e do not have complex multiplication.Modular form 270802.2.a.e
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.