Properties

Label 162240.ea
Number of curves $4$
Conductor $162240$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("ea1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 162240.ea

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
162240.ea1 162240de4 \([0, -1, 0, -5624545, -5132401535]\) \(31103978031362/195\) \(123368604303360\) \([2]\) \(4128768\) \(2.3094\)  
162240.ea2 162240de3 \([0, -1, 0, -486945, -12702015]\) \(20183398562/11567205\) \(7318102238671011840\) \([2]\) \(4128768\) \(2.3094\)  
162240.ea3 162240de2 \([0, -1, 0, -351745, -80004575]\) \(15214885924/38025\) \(12028438919577600\) \([2, 2]\) \(2064384\) \(1.9629\)  
162240.ea4 162240de1 \([0, -1, 0, -13745, -2196975]\) \(-3631696/24375\) \(-1927634442240000\) \([2]\) \(1032192\) \(1.6163\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 162240.ea have rank \(1\).

Complex multiplication

The elliptic curves in class 162240.ea do not have complex multiplication.

Modular form 162240.2.a.ea

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.