Properties

Label 258570eg
Number of curves 4
Conductor 258570
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("258570.eg1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 258570eg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
258570.eg4 258570eg1 [1, -1, 1, 2403148, -243740649] [4] 15482880 \(\Gamma_0(N)\)-optimal
258570.eg3 258570eg2 [1, -1, 1, -9764852, -1956995049] [2, 2] 30965760  
258570.eg2 258570eg3 [1, -1, 1, -97678652, 369672220311] [2] 61931520  
258570.eg1 258570eg4 [1, -1, 1, -116539052, -483380508009] [2] 61931520  

Rank

sage: E.rank()
 

The elliptic curves in class 258570eg have rank \(1\).

Modular form 258570.2.a.eg

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

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 Cremona 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 Cremona labels.