Properties

Label 397488ir
Number of curves $4$
Conductor $397488$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("ir1")
 
E.isogeny_class()
 

Elliptic curves in class 397488ir

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
397488.ir3 397488ir1 \([0, 1, 0, -930232, -309720748]\) \(38272753/4368\) \(10159935049380200448\) \([2]\) \(9289728\) \(2.3794\) \(\Gamma_0(N)\)-optimal
397488.ir2 397488ir2 \([0, 1, 0, -3580152, 2277661140]\) \(2181825073/298116\) \(693415567120198680576\) \([2, 2]\) \(18579456\) \(2.7260\)  
397488.ir1 397488ir3 \([0, 1, 0, -55253592, 158062748052]\) \(8020417344913/187278\) \(435607215242176094208\) \([2]\) \(37158912\) \(3.0725\)  
397488.ir4 397488ir4 \([0, 1, 0, 5694568, 12127413780]\) \(8780064047/32388174\) \(-75334648399273013796864\) \([2]\) \(37158912\) \(3.0725\)  

Rank

sage: E.rank()
 

The elliptic curves in class 397488ir have rank \(0\).

Complex multiplication

The elliptic curves in class 397488ir do not have complex multiplication.

Modular form 397488.2.a.ir

sage: E.q_eigenform(10)
 
\(q + q^{3} + 2 q^{5} + q^{9} - 4 q^{11} + 2 q^{15} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.