Properties

Label 236992.bp
Number of curves $4$
Conductor $236992$
CM no
Rank $1$
Graph

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

Elliptic curves in class 236992.bp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
236992.bp1 236992bp3 \([0, 0, 0, -632684, 193698640]\) \(1443468546/7\) \(135823520301056\) \([2]\) \(1622016\) \(1.9123\)  
236992.bp2 236992bp4 \([0, 0, 0, -124844, -13432368]\) \(11090466/2401\) \(46587467463262208\) \([2]\) \(1622016\) \(1.9123\)  
236992.bp3 236992bp2 \([0, 0, 0, -40204, 2920080]\) \(740772/49\) \(475382321053696\) \([2, 2]\) \(811008\) \(1.5657\)  
236992.bp4 236992bp1 \([0, 0, 0, 2116, 194672]\) \(432/7\) \(-16977940037632\) \([2]\) \(405504\) \(1.2191\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 236992.bp have rank \(1\).

Complex multiplication

The elliptic curves in class 236992.bp do not have complex multiplication.

Modular form 236992.2.a.bp

sage: E.q_eigenform(10)
 
\(q + 2q^{5} - q^{7} - 3q^{9} + 4q^{11} - 2q^{13} + 6q^{17} - 8q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.