Show commands:
SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 142956.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
142956.p1 | 142956k2 | \([0, 0, 0, -4198791, 3311484046]\) | \(932410994128/29403\) | \(258155120246360832\) | \([2]\) | \(3421440\) | \(2.4369\) | |
142956.p2 | 142956k1 | \([0, 0, 0, -251256, 56346685]\) | \(-3196715008/649539\) | \(-356430080794691376\) | \([2]\) | \(1710720\) | \(2.0903\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 142956.p have rank \(1\).
Complex multiplication
The elliptic curves in class 142956.p do not have complex multiplication.Modular form 142956.2.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 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.