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Properties

Label 463680.bm
Number of curves $2$
Conductor $463680$
CM no
Rank $1$
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 463680.bm have rank \(1\).

Complex multiplication

The elliptic curves in class 463680.bm do not have complex multiplication.

Modular form 463680.2.a.bm

Copy content sage:E.q_eigenform(10)
 
\(q - q^{5} - q^{7} - 4 q^{13} - 4 q^{17} + 2 q^{19} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.

Elliptic curves in class 463680.bm

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
463680.bm1 463680bm2 \([0, 0, 0, -8991468, 10376329392]\) \(420676324562824569/56350000000\) \(10768652697600000000\) \([2]\) \(16515072\) \(2.6710\) \(\Gamma_0(N)\)-optimal*
463680.bm2 463680bm1 \([0, 0, 0, -512748, 191690928]\) \(-78013216986489/37918720000\) \(-7246380238110720000\) \([2]\) \(8257536\) \(2.3244\) \(\Gamma_0(N)\)-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 463680.bm1.