Properties

Label 333270.bj
Number of curves $4$
Conductor $333270$
CM no
Rank $2$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("bj1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 333270.bj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.bj1 333270bj4 \([1, -1, 0, -2296516059, -41455596567987]\) \(12411881707829361287041/303132494474220600\) \(32713501973819270061169668600\) \([2]\) \(525533184\) \(4.2555\)  
333270.bj2 333270bj2 \([1, -1, 0, -282613059, 1806613478613]\) \(23131609187144855041/322060536000000\) \(34756181446002271416000000\) \([2]\) \(175177728\) \(3.7062\)  
333270.bj3 333270bj1 \([1, -1, 0, -2285379, 75702185685]\) \(-12232183057921/22933241856000\) \(-2474913334591823118336000\) \([2]\) \(87588864\) \(3.3597\) \(\Gamma_0(N)\)-optimal
333270.bj4 333270bj3 \([1, -1, 0, 20567421, -2043396823275]\) \(8915971454369279/16719623332762560\) \(-1804351037477962680331047360\) \([2]\) \(262766592\) \(3.9090\)  

Rank

sage: E.rank()
 

The elliptic curves in class 333270.bj have rank \(2\).

Complex multiplication

The elliptic curves in class 333270.bj do not have complex multiplication.

Modular form 333270.2.a.bj

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8} - q^{10} - 6q^{11} - 4q^{13} + q^{14} + q^{16} - 6q^{17} + 4q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.