Show commands:
SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 1216.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1216.o1 | 1216d3 | \([0, -1, 0, -3077, -64681]\) | \(-50357871050752/19\) | \(-1216\) | \([]\) | \(432\) | \(0.38001\) | |
1216.o2 | 1216d2 | \([0, -1, 0, -37, -81]\) | \(-89915392/6859\) | \(-438976\) | \([]\) | \(144\) | \(-0.16929\) | |
1216.o3 | 1216d1 | \([0, -1, 0, 3, -1]\) | \(32768/19\) | \(-1216\) | \([]\) | \(48\) | \(-0.71860\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1216.o have rank \(1\).
Complex multiplication
The elliptic curves in class 1216.o do not have complex multiplication.Modular form 1216.2.a.o
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.