↑

Properties

Label 462462.fo
Number of curves $3$
Conductor $462462$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("fo1")
 
E.isogeny_class()
 

Elliptic curves in class 462462.fo

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
462462.fo1 462462fo3 \([1, 1, 1, -154489112, -739162250023]\) \(-1956469094246217097/36641439744\) \(-7636896081332158857216\) \([]\) \(125971200\) \(3.3234\)  
462462.fo2 462462fo2 \([1, 1, 1, -720497, -2254488433]\) \(-198461344537/10417365504\) \(-2171212112600725049856\) \([]\) \(41990400\) \(2.7741\)  
462462.fo3 462462fo1 \([1, 1, 1, 79918, 82723367]\) \(270840023/14329224\) \(-2986530970908420936\) \([]\) \(13996800\) \(2.2248\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 462462.fo1.

Rank

sage: E.rank()
 

The elliptic curves in class 462462.fo have rank \(1\).

Complex multiplication

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

Modular form 462462.2.a.fo

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - 3 q^{5} - q^{6} + q^{8} + q^{9} - 3 q^{10} - q^{12} + q^{13} + 3 q^{15} + q^{16} - 3 q^{17} + q^{18} - 7 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.