Properties

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

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

Elliptic curves in class 259920ff

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
259920.ff3 259920ff1 \([0, 0, 0, -1612587, -783476134]\) \(3301293169/22800\) \(3202904052847411200\) \([2]\) \(4423680\) \(2.3844\) \(\Gamma_0(N)\)-optimal
259920.ff2 259920ff2 \([0, 0, 0, -2652267, 350398874]\) \(14688124849/8122500\) \(1141034568826890240000\) \([2, 2]\) \(8847360\) \(2.7310\)  
259920.ff1 259920ff3 \([0, 0, 0, -32283147, 70498544186]\) \(26487576322129/44531250\) \(6255671978217600000000\) \([4]\) \(17694720\) \(3.0776\)  
259920.ff4 259920ff4 \([0, 0, 0, 10343733, 2770254074]\) \(871257511151/527800050\) \(-74144426282371327795200\) \([2]\) \(17694720\) \(3.0776\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 259920ff do not have complex multiplication.

Modular form 259920.2.a.ff

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