Properties

Label 324870fo
Number of curves $4$
Conductor $324870$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

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

Elliptic curves in class 324870fo

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
324870.fo4 324870fo1 \([1, 0, 0, 45814, -2965740]\) \(90391899763439/84690294000\) \(-9963728398806000\) \([2]\) \(2433024\) \(1.7570\) \(\Gamma_0(N)\)-optimal
324870.fo3 324870fo2 \([1, 0, 0, -237406, -26812864]\) \(12577973014374481/4642947562500\) \(546238137780562500\) \([2, 2]\) \(4866048\) \(2.1036\)  
324870.fo2 324870fo3 \([1, 0, 0, -1645176, 792790830]\) \(4185743240664514801/113629394531250\) \(13368384637207031250\) \([2]\) \(9732096\) \(2.4502\)  
324870.fo1 324870fo4 \([1, 0, 0, -3361156, -2371499614]\) \(35694515311673154481/10400566692750\) \(1223616270835344750\) \([2]\) \(9732096\) \(2.4502\)  

Rank

sage: E.rank()
 

The elliptic curves in class 324870fo have rank \(1\).

Complex multiplication

The elliptic curves in class 324870fo do not have complex multiplication.

Modular form 324870.2.a.fo

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} - q^{10} + 4 q^{11} + q^{12} + q^{13} - q^{15} + q^{16} - q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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.