Properties

Label 129360.fj
Number of curves $6$
Conductor $129360$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("fj1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 129360.fj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
129360.fj1 129360gk6 \([0, 1, 0, -2390572816, -44989251846316]\) \(3135316978843283198764801/571725\) \(275508734054400\) \([2]\) \(23592960\) \(3.5652\)  
129360.fj2 129360gk4 \([0, 1, 0, -149410816, -702994261516]\) \(765458482133960722801/326869475625\) \(157515230977251840000\) \([2, 2]\) \(11796480\) \(3.2187\)  
129360.fj3 129360gk5 \([0, 1, 0, -148669936, -710310303340]\) \(-754127868744065783521/15825714261328125\) \(-7626258256408545600000000\) \([2]\) \(23592960\) \(3.5652\)  
129360.fj4 129360gk3 \([0, 1, 0, -19948896, 18078514740]\) \(1821931919215868881/761147600816295\) \(366789648746235045703680\) \([2]\) \(11796480\) \(3.2187\)  
129360.fj5 129360gk2 \([0, 1, 0, -9384496, -10872167020]\) \(189674274234120481/3859869269025\) \(1860033575450715033600\) \([2, 2]\) \(5898240\) \(2.8721\)  
129360.fj6 129360gk1 \([0, 1, 0, 27424, -507760716]\) \(4733169839/231139696095\) \(-111383978417687162880\) \([2]\) \(2949120\) \(2.5255\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 129360.2.a.fj

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.