Properties

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

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 1216.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1216.b1 1216q3 \([0, 1, 0, -3077, 64681]\) \(-50357871050752/19\) \(-1216\) \([]\) \(432\) \(0.38001\)  
1216.b2 1216q2 \([0, 1, 0, -37, 81]\) \(-89915392/6859\) \(-438976\) \([]\) \(144\) \(-0.16929\)  
1216.b3 1216q1 \([0, 1, 0, 3, 1]\) \(32768/19\) \(-1216\) \([]\) \(48\) \(-0.71860\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 1216.b have rank \(1\).

Complex multiplication

The elliptic curves in class 1216.b do not have complex multiplication.

Modular form 1216.2.a.b

sage: E.q_eigenform(10)
 
\(q - 2 q^{3} - 3 q^{5} + q^{7} + q^{9} + 3 q^{11} + 4 q^{13} + 6 q^{15} - 3 q^{17} + q^{19} + O(q^{20})\) Copy content 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.