Show commands:
SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 172788.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
172788.p1 | 172788bf1 | \([0, -1, 0, -40817, 3184590]\) | \(265327034368/297381\) | \(8429257307856\) | \([2]\) | \(518400\) | \(1.3943\) | \(\Gamma_0(N)\)-optimal |
172788.p2 | 172788bf2 | \([0, -1, 0, -30532, 4817848]\) | \(-6940769488/18000297\) | \(-8163487783325952\) | \([2]\) | \(1036800\) | \(1.7409\) |
Rank
sage: E.rank()
The elliptic curves in class 172788.p have rank \(1\).
Complex multiplication
The elliptic curves in class 172788.p do not have complex multiplication.Modular form 172788.2.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.