Properties

Label 406560.bp
Number of curves $4$
Conductor $406560$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

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

Elliptic curves in class 406560.bp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
406560.bp1 406560bp2 \([0, -1, 0, -38420081, 91674005601]\) \(864335783029582144/59535\) \(432004645416960\) \([2]\) \(19660800\) \(2.7092\) \(\Gamma_0(N)\)-optimal*
406560.bp2 406560bp4 \([0, -1, 0, -2697856, 1057037236]\) \(2394165105226952/854262178245\) \(774849310081991907840\) \([2]\) \(19660800\) \(2.7092\)  
406560.bp3 406560bp1 \([0, -1, 0, -2401406, 1432817256]\) \(13507798771700416/3544416225\) \(401865571326542400\) \([2, 2]\) \(9830400\) \(2.3626\) \(\Gamma_0(N)\)-optimal*
406560.bp4 406560bp3 \([0, -1, 0, -2107376, 1796473560]\) \(-1141100604753992/875529151875\) \(-794139289510311360000\) \([2]\) \(19660800\) \(2.7092\)  
*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 406560.bp1.

Rank

sage: E.rank()
 

The elliptic curves in class 406560.bp have rank \(0\).

Complex multiplication

The elliptic curves in class 406560.bp do not have complex multiplication.

Modular form 406560.2.a.bp

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{7} + q^{9} + 6 q^{13} + q^{15} - 6 q^{17} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.