Properties

Label 463680bs
Number of curves $4$
Conductor $463680$
CM no
Rank $2$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("bs1")
 
E.isogeny_class()
 

Elliptic curves in class 463680bs

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
463680.bs4 463680bs1 \([0, 0, 0, 792, -20088]\) \(73598976/276115\) \(-206118743040\) \([2]\) \(327680\) \(0.85380\) \(\Gamma_0(N)\)-optimal*
463680.bs3 463680bs2 \([0, 0, 0, -8028, -242352]\) \(4790692944/648025\) \(7739969126400\) \([2, 2]\) \(655360\) \(1.2004\) \(\Gamma_0(N)\)-optimal*
463680.bs2 463680bs3 \([0, 0, 0, -33228, 2086128]\) \(84923690436/9794435\) \(467936419184640\) \([2]\) \(1310720\) \(1.5469\) \(\Gamma_0(N)\)-optimal*
463680.bs1 463680bs4 \([0, 0, 0, -123948, -16795728]\) \(4407931365156/100625\) \(4807434240000\) \([2]\) \(1310720\) \(1.5469\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 463680bs1.

Rank

sage: E.rank()
 

The elliptic curves in class 463680bs have rank \(2\).

Complex multiplication

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

Modular form 463680.2.a.bs

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.