Properties

Label 89280ca
Number of curves $6$
Conductor $89280$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("89280.ed1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 89280ca

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
89280.ed6 89280ca1 [0, 0, 0, 34548, 14003696] [2] 786432 \(\Gamma_0(N)\)-optimal
89280.ed5 89280ca2 [0, 0, 0, -702732, 214248944] [2, 2] 1572864  
89280.ed4 89280ca3 [0, 0, 0, -2131212, -933677584] [2] 3145728  
89280.ed2 89280ca4 [0, 0, 0, -11070732, 14177871344] [2, 2] 3145728  
89280.ed3 89280ca5 [0, 0, 0, -10897932, 14641873904] [2] 6291456  
89280.ed1 89280ca6 [0, 0, 0, -177131532, 907385702384] [2] 6291456  

Rank

sage: E.rank()
 

The elliptic curves in class 89280ca have rank \(1\).

Modular form 89280.2.a.ed

sage: E.q_eigenform(10)
 
\( q + q^{5} - 4q^{11} - 6q^{13} - 2q^{17} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.