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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 121968.by
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121968.by1 | 121968x1 | \([0, 0, 0, -726, 6655]\) | \(55296/7\) | \(5357200464\) | \([2]\) | \(69120\) | \(0.59603\) | \(\Gamma_0(N)\)-optimal |
121968.by2 | 121968x2 | \([0, 0, 0, 1089, 34606]\) | \(11664/49\) | \(-600006451968\) | \([2]\) | \(138240\) | \(0.94260\) |
Rank
sage: E.rank()
The elliptic curves in class 121968.by have rank \(1\).
Complex multiplication
The elliptic curves in class 121968.by do not have complex multiplication.Modular form 121968.2.a.by
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.