Show commands:
SageMath
E = EllipticCurve("ha1")
E.isogeny_class()
Elliptic curves in class 221760ha
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
221760.dt1 | 221760ha1 | \([0, 0, 0, -134028, 17461008]\) | \(51603494067/4336640\) | \(22376109865697280\) | \([2]\) | \(1474560\) | \(1.8790\) | \(\Gamma_0(N)\)-optimal |
221760.dt2 | 221760ha2 | \([0, 0, 0, 142452, 80166672]\) | \(61958108493/573927200\) | \(-2961338290038374400\) | \([2]\) | \(2949120\) | \(2.2256\) |
Rank
sage: E.rank()
The elliptic curves in class 221760ha have rank \(1\).
Complex multiplication
The elliptic curves in class 221760ha do not have complex multiplication.Modular form 221760.2.a.ha
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.