↑

Properties

Label 462462.fz
Number of curves $2$
Conductor $462462$
CM no
Rank $2$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("fz1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 462462.fz have rank \(2\).

Complex multiplication

The elliptic curves in class 462462.fz do not have complex multiplication.

Modular form 462462.2.a.fz

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - 2 q^{5} - q^{6} + q^{8} + q^{9} - 2 q^{10} - q^{12} + q^{13} + 2 q^{15} + q^{16} - 2 q^{17} + q^{18} + 2 q^{19} + O(q^{20})\) 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 462462.fz

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
462462.fz1 462462fz2 \([1, 1, 1, -32194, 1718135]\) \(23565848363/5366088\) \(840280114746072\) \([2]\) \(2654208\) \(1.5763\) \(\Gamma_0(N)\)-optimal*
462462.fz2 462462fz1 \([1, 1, 1, -10634, -403369]\) \(849278123/52416\) \(8207864368704\) \([2]\) \(1327104\) \(1.2297\) \(\Gamma_0(N)\)-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 462462.fz1.