Properties

Label 114240.hh
Number of curves $4$
Conductor $114240$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("hh1")
 
E.isogeny_class()
 

Elliptic curves in class 114240.hh

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
114240.hh1 114240ea4 \([0, 1, 0, -64961, 4896159]\) \(115650783909361/27072079335\) \(7096783165194240\) \([2]\) \(786432\) \(1.7534\)  
114240.hh2 114240ea2 \([0, 1, 0, -21761, -1177761]\) \(4347507044161/258084225\) \(67655231078400\) \([2, 2]\) \(393216\) \(1.4068\)  
114240.hh3 114240ea1 \([0, 1, 0, -21441, -1215585]\) \(4158523459441/16065\) \(4211343360\) \([2]\) \(196608\) \(1.0602\) \(\Gamma_0(N)\)-optimal
114240.hh4 114240ea3 \([0, 1, 0, 16319, -4825825]\) \(1833318007919/39525924375\) \(-10361483919360000\) \([2]\) \(786432\) \(1.7534\)  

Rank

sage: E.rank()
 

The elliptic curves in class 114240.hh have rank \(2\).

Complex multiplication

The elliptic curves in class 114240.hh do not have complex multiplication.

Modular form 114240.2.a.hh

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{5} + q^{7} + q^{9} - 4 q^{11} - 2 q^{13} - q^{15} + q^{17} - 8 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.