Properties

Label 324870bn
Number of curves 4
Conductor 324870
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("324870.bn1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 324870bn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
324870.bn4 324870bn1 [1, 0, 1, -265949, 306296] [2] 5308416 \(\Gamma_0(N)\)-optimal
324870.bn2 324870bn2 [1, 0, 1, -2915869, -1910816008] [2, 2] 10616832  
324870.bn3 324870bn3 [1, 0, 1, -1616389, -3622491064] [2] 21233664  
324870.bn1 324870bn4 [1, 0, 1, -46614069, -122500368728] [2] 21233664  

Rank

sage: E.rank()
 

The elliptic curves in class 324870bn have rank \(1\).

Modular form 324870.2.a.bn

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{8} + q^{9} + q^{10} - 4q^{11} + q^{12} - q^{13} - q^{15} + q^{16} - q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

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.