Show commands:
SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 271950p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
271950.p1 | 271950p1 | \([1, 1, 0, -13500, -1144800]\) | \(-3700897225/5482512\) | \(-403132533930000\) | \([]\) | \(1824768\) | \(1.4943\) | \(\Gamma_0(N)\)-optimal |
271950.p2 | 271950p2 | \([1, 1, 0, 115125, 22753725]\) | \(2294872120775/4356968448\) | \(-320370613086720000\) | \([]\) | \(5474304\) | \(2.0436\) |
Rank
sage: E.rank()
The elliptic curves in class 271950p have rank \(2\).
Complex multiplication
The elliptic curves in class 271950p do not have complex multiplication.Modular form 271950.2.a.p
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.