Properties

Label 259920ff
Number of curves $4$
Conductor $259920$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("259920.ff1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 259920ff

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
259920.ff3 259920ff1 [0, 0, 0, -1612587, -783476134] [2] 4423680 \(\Gamma_0(N)\)-optimal
259920.ff2 259920ff2 [0, 0, 0, -2652267, 350398874] [2, 2] 8847360  
259920.ff1 259920ff3 [0, 0, 0, -32283147, 70498544186] [4] 17694720  
259920.ff4 259920ff4 [0, 0, 0, 10343733, 2770254074] [2] 17694720  

Rank

sage: E.rank()
 

The elliptic curves in class 259920ff have rank \(1\).

Modular form 259920.2.a.ff

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

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.