Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 6930.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6930.e1 | 6930b2 | \([1, -1, 0, -1275, 17845]\) | \(8493409990827/474320\) | \(12806640\) | \([2]\) | \(3072\) | \(0.42885\) | |
6930.e2 | 6930b1 | \([1, -1, 0, -75, 325]\) | \(-1740992427/492800\) | \(-13305600\) | \([2]\) | \(1536\) | \(0.082274\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6930.e have rank \(1\).
Complex multiplication
The elliptic curves in class 6930.e do not have complex multiplication.Modular form 6930.2.a.e
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.