Properties

Label 291312bp
Number of curves $6$
Conductor $291312$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("291312.bp1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 291312bp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
291312.bp5 291312bp1 [0, 0, 0, -2923275171, 60758772299426] [2] 212336640 \(\Gamma_0(N)\)-optimal
291312.bp4 291312bp2 [0, 0, 0, -3775570851, 22441092658850] [2, 2] 424673280  
291312.bp6 291312bp3 [0, 0, 0, 14562103389, 176503229018786] [2] 849346560  
291312.bp2 291312bp4 [0, 0, 0, -35749975971, -2583952540697950] [2, 2] 849346560  
291312.bp3 291312bp5 [0, 0, 0, -12177008931, -5940634611545566] [2] 1698693120  
291312.bp1 291312bp6 [0, 0, 0, -570913424931, -166036463004685534] [2] 1698693120  

Rank

sage: E.rank()
 

The elliptic curves in class 291312bp have rank \(0\).

Modular form 291312.2.a.bp

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.