Properties

Label 129360fo
Number of curves $4$
Conductor $129360$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("fo1")
 
E.isogeny_class()
 

Elliptic curves in class 129360fo

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
129360.dh3 129360fo1 \([0, -1, 0, -392800, -82146560]\) \(13908844989649/1980372240\) \(954322180766760960\) \([2]\) \(1769472\) \(2.1757\) \(\Gamma_0(N)\)-optimal
129360.dh2 129360fo2 \([0, -1, 0, -1662880, 743913472]\) \(1055257664218129/115307784900\) \(55565703519027609600\) \([2, 2]\) \(3538944\) \(2.5223\)  
129360.dh4 129360fo3 \([0, -1, 0, 2217920, 3696426112]\) \(2503876820718671/13702874328990\) \(-6603282276070787112960\) \([2]\) \(7077888\) \(2.8689\)  
129360.dh1 129360fo4 \([0, -1, 0, -25864960, 50638921600]\) \(3971101377248209009/56495958750\) \(27224854736808960000\) \([4]\) \(7077888\) \(2.8689\)  

Rank

sage: E.rank()
 

The elliptic curves in class 129360fo have rank \(1\).

Complex multiplication

The elliptic curves in class 129360fo do not have complex multiplication.

Modular form 129360.2.a.fo

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} + q^{9} + q^{11} - 2 q^{13} - q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 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.