Show commands:
SageMath
E = EllipticCurve("ez1")
E.isogeny_class()
Elliptic curves in class 68544.ez
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68544.ez1 | 68544cr2 | \([0, 0, 0, -74028, 1472560]\) | \(234770924809/130960928\) | \(25027023080521728\) | \([2]\) | \(737280\) | \(1.8369\) | |
68544.ez2 | 68544cr1 | \([0, 0, 0, 18132, 182320]\) | \(3449795831/2071552\) | \(-395879752138752\) | \([2]\) | \(368640\) | \(1.4903\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 68544.ez have rank \(0\).
Complex multiplication
The elliptic curves in class 68544.ez do not have complex multiplication.Modular form 68544.2.a.ez
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.