Properties

Label 229320.i
Number of curves $4$
Conductor $229320$
CM no
Rank $2$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("i1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 229320.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
229320.i1 229320bf4 \([0, 0, 0, -730443, -239377642]\) \(490757540836/2142075\) \(188126682768460800\) \([2]\) \(3538944\) \(2.1676\)  
229320.i2 229320bf2 \([0, 0, 0, -68943, 482258]\) \(1650587344/950625\) \(20872043206560000\) \([2, 2]\) \(1769472\) \(1.8210\)  
229320.i3 229320bf1 \([0, 0, 0, -49098, 4177397]\) \(9538484224/26325\) \(36124690165200\) \([2]\) \(884736\) \(1.4744\) \(\Gamma_0(N)\)-optimal
229320.i4 229320bf3 \([0, 0, 0, 275037, 3853262]\) \(26198797244/15234375\) \(-1337951487600000000\) \([2]\) \(3538944\) \(2.1676\)  

Rank

sage: E.rank()
 

The elliptic curves in class 229320.i have rank \(2\).

Complex multiplication

The elliptic curves in class 229320.i do not have complex multiplication.

Modular form 229320.2.a.i

sage: E.q_eigenform(10)
 
\(q - q^{5} - 4 q^{11} - q^{13} - 6 q^{17} + O(q^{20})\)  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 LMFDB 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 LMFDB labels.