↑

Properties

Label 431970.eg
Number of curves $2$
Conductor $431970$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("eg1")
 
E.isogeny_class()
 

Elliptic curves in class 431970.eg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
431970.eg1 431970eg2 \([1, 1, 1, -4559706, 3741039303]\) \(5918043195362419129/8515734343200\) \(15086142848773735200\) \([2]\) \(21504000\) \(2.5825\) \(\Gamma_0(N)\)-optimal*
431970.eg2 431970eg1 \([1, 1, 1, -203706, 92453703]\) \(-527690404915129/1782829440000\) \(-3158391105555840000\) \([2]\) \(10752000\) \(2.2359\) \(\Gamma_0(N)\)-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 431970.eg1.

Rank

sage: E.rank()
 

The elliptic curves in class 431970.eg have rank \(1\).

Complex multiplication

The elliptic curves in class 431970.eg do not have complex multiplication.

Modular form 431970.2.a.eg

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{7} + q^{8} + q^{9} - q^{10} - q^{12} - 4 q^{13} + q^{14} + q^{15} + q^{16} + q^{17} + q^{18} + 6 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.