Show commands:
SageMath
E = EllipticCurve("hw1")
E.isogeny_class()
Elliptic curves in class 129600.hw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
129600.hw1 | 129600eh2 | \([0, 0, 0, -3242700, 2257254000]\) | \(-15590912409/78125\) | \(-18895680000000000000\) | \([]\) | \(3096576\) | \(2.5461\) | |
129600.hw2 | 129600eh1 | \([0, 0, 0, -2700, -1674000]\) | \(-9/5\) | \(-1209323520000000\) | \([]\) | \(442368\) | \(1.5732\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 129600.hw have rank \(0\).
Complex multiplication
The elliptic curves in class 129600.hw do not have complex multiplication.Modular form 129600.2.a.hw
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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.