Show commands:
SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 294350cm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
294350.cm2 | 294350cm1 | \([1, 0, 0, 3767, 74977]\) | \(397535/392\) | \(-5829268545800\) | \([]\) | \(580608\) | \(1.1365\) | \(\Gamma_0(N)\)-optimal |
294350.cm1 | 294350cm2 | \([1, 0, 0, -38283, -4054333]\) | \(-417267265/235298\) | \(-3499018444616450\) | \([]\) | \(1741824\) | \(1.6858\) |
Rank
sage: E.rank()
The elliptic curves in class 294350cm have rank \(0\).
Complex multiplication
The elliptic curves in class 294350cm do not have complex multiplication.Modular form 294350.2.a.cm
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.