Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 3648.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3648.a1 | 3648bc1 | \([0, -1, 0, -65, -159]\) | \(470596/57\) | \(3735552\) | \([2]\) | \(1280\) | \(-0.0083494\) | \(\Gamma_0(N)\)-optimal |
3648.a2 | 3648bc2 | \([0, -1, 0, 95, -959]\) | \(715822/3249\) | \(-425852928\) | \([2]\) | \(2560\) | \(0.33822\) |
Rank
sage: E.rank()
The elliptic curves in class 3648.a have rank \(1\).
Complex multiplication
The elliptic curves in class 3648.a do not have complex multiplication.Modular form 3648.2.a.a
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.