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SageMath
E = EllipticCurve("cu1")
E.isogeny_class()
Elliptic curves in class 121680cu
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121680.eb2 | 121680cu1 | \([0, 0, 0, 32448, 2917616]\) | \(7077888/10985\) | \(-5863863973294080\) | \([]\) | \(580608\) | \(1.7109\) | \(\Gamma_0(N)\)-optimal |
121680.eb1 | 121680cu2 | \([0, 0, 0, -1022112, 399572784]\) | \(-303464448/1625\) | \(-632360478776832000\) | \([]\) | \(1741824\) | \(2.2602\) |
Rank
sage: E.rank()
The elliptic curves in class 121680cu have rank \(1\).
Complex multiplication
The elliptic curves in class 121680cu do not have complex multiplication.Modular form 121680.2.a.cu
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.