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SageMath
E = EllipticCurve("dp1")
E.isogeny_class()
Elliptic curves in class 205350dp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
205350.u2 | 205350dp1 | \([1, 1, 0, -65740, 5098000]\) | \(97972181/21312\) | \(6835095153576000\) | \([2]\) | \(2101248\) | \(1.7522\) | \(\Gamma_0(N)\)-optimal |
205350.u1 | 205350dp2 | \([1, 1, 0, -339540, -71839800]\) | \(13498272341/887112\) | \(284510835767601000\) | \([2]\) | \(4202496\) | \(2.0988\) |
Rank
sage: E.rank()
The elliptic curves in class 205350dp have rank \(0\).
Complex multiplication
The elliptic curves in class 205350dp do not have complex multiplication.Modular form 205350.2.a.dp
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.