Show commands:
SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 83391.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
83391.i1 | 83391g2 | \([0, -1, 1, -1001173, 387169956]\) | \(-2359010787328000/8925676683\) | \(-419916323072892723\) | \([]\) | \(1088640\) | \(2.2409\) | |
83391.i2 | 83391g1 | \([0, -1, 1, 27677, 2771019]\) | \(49836032000/99819027\) | \(-4696074065777787\) | \([]\) | \(362880\) | \(1.6916\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 83391.i have rank \(1\).
Complex multiplication
The elliptic curves in class 83391.i do not have complex multiplication.Modular form 83391.2.a.i
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.