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Properties

Label 466578bm
Number of curves $2$
Conductor $466578$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bm1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 466578bm have rank \(0\).

Complex multiplication

The elliptic curves in class 466578bm do not have complex multiplication.

Modular form 466578.2.a.bm

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{8} - 4 q^{11} + 2 q^{13} + q^{16} - 2 q^{17} - 6 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 466578bm

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
466578.bm2 466578bm1 \([1, -1, 0, -99852552, -388002661376]\) \(-25282750375/304704\) \(-1326951531451662621317568\) \([2]\) \(68124672\) \(3.4404\) \(\Gamma_0(N)\)-optimal*
466578.bm1 466578bm2 \([1, -1, 0, -1602233712, -24684811257128]\) \(104453838382375/14904\) \(64905237951440019520968\) \([2]\) \(136249344\) \(3.7870\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 466578bm1.