Properties

Label 468270.bz
Number of curves $2$
Conductor $468270$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bz1")
 
E.isogeny_class()
 

Elliptic curves in class 468270.bz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
468270.bz1 468270bz2 \([1, -1, 0, -212559, -25957387]\) \(30459021867/9245000\) \(322369777081935000\) \([2]\) \(6451200\) \(2.0646\)  
468270.bz2 468270bz1 \([1, -1, 0, -81879, 8725085]\) \(1740992427/68800\) \(2399030899214400\) \([2]\) \(3225600\) \(1.7180\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 468270.bz1.

Rank

sage: E.rank()
 

The elliptic curves in class 468270.bz have rank \(0\).

Complex multiplication

The elliptic curves in class 468270.bz do not have complex multiplication.

Modular form 468270.2.a.bz

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.