Show commands:
SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 321346.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
321346.b1 | 321346b2 | \([1, 0, 0, -1547577, 740872825]\) | \(409902464594359487449873/7936575003324416\) | \(7936575003324416\) | \([2]\) | \(4521984\) | \(2.1734\) | |
321346.b2 | 321346b1 | \([1, 0, 0, -93497, 12378745]\) | \(-90389301567546411793/13971531281465344\) | \(-13971531281465344\) | \([2]\) | \(2260992\) | \(1.8269\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 321346.b have rank \(2\).
Complex multiplication
The elliptic curves in class 321346.b do not have complex multiplication.Modular form 321346.2.a.b
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.