Properties

Label 288990.ea
Number of curves $4$
Conductor $288990$
CM no
Rank $1$
Graph

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

Elliptic curves in class 288990.ea

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
288990.ea1 288990ea4 \([1, -1, 1, -704574383, -4596367959673]\) \(10993009831928446009969/3767761230468750000\) \(13257786322649597167968750000\) \([2]\) \(298598400\) \(4.0981\)  
288990.ea2 288990ea2 \([1, -1, 1, -631201343, -6103637247769]\) \(7903870428425797297009/886464000000\) \(3119239669351104000000\) \([2]\) \(99532800\) \(3.5488\)  
288990.ea3 288990ea1 \([1, -1, 1, -39349823, -95870838553]\) \(-1914980734749238129/20440940544000\) \(-71926432008171945984000\) \([2]\) \(49766400\) \(3.2023\) \(\Gamma_0(N)\)-optimal
288990.ea4 288990ea3 \([1, -1, 1, 130028737, -499134322969]\) \(69096190760262356111/70568821500000000\) \(-248313600374247661500000000\) \([2]\) \(149299200\) \(3.7516\)  

Rank

sage: E.rank()
 

The elliptic curves in class 288990.ea have rank \(1\).

Complex multiplication

The elliptic curves in class 288990.ea do not have complex multiplication.

Modular form 288990.2.a.ea

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{5} - 2 q^{7} + q^{8} - q^{10} + 6 q^{11} - 2 q^{14} + q^{16} + 6 q^{17} - q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.