Properties

Label 1216.o
Number of curves $3$
Conductor $1216$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("o1")
 
sage: 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)
 
\(q + 2q^{3} - 3q^{5} - q^{7} + q^{9} - 3q^{11} + 4q^{13} - 6q^{15} - 3q^{17} - q^{19} + O(q^{20})\)  Toggle raw display

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.