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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 189630.dd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 189630.dd1 | 189630bz2 | \([1, -1, 1, -86078, -6681419]\) | \(30459021867/9245000\) | \(21408510293415000\) | \([2]\) | \(1658880\) | \(1.8386\) | |
| 189630.dd2 | 189630bz1 | \([1, -1, 1, -33158, 2251477]\) | \(1740992427/68800\) | \(159319146369600\) | \([2]\) | \(829440\) | \(1.4920\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 189630.dd have rank \(1\).
Complex multiplication
The elliptic curves in class 189630.dd do not have complex multiplication.Modular form 189630.2.a.dd
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.
