Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1216.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1216.e1 1216b3 \([0, -1, 0, -5473, -1251871]\) \(-69173457625/2550136832\) \(-668503069687808\) \([]\) \(3456\) \(1.5249\)  
1216.e2 1216b1 \([0, -1, 0, -993, 12385]\) \(-413493625/152\) \(-39845888\) \([]\) \(384\) \(0.42628\) \(\Gamma_0(N)\)-optimal
1216.e3 1216b2 \([0, -1, 0, 607, 45601]\) \(94196375/3511808\) \(-920599396352\) \([]\) \(1152\) \(0.97558\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 1216.2.a.e

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

\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.