Properties

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

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 324870.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
324870.k1 324870k3 \([1, 1, 0, -25802298, 50436290388]\) \(16147699491119193630361/273400196160\) \(32165259678027840\) \([2]\) \(17915904\) \(2.7099\)  
324870.k2 324870k4 \([1, 1, 0, -25776818, 50540906172]\) \(-16099908724056356461081/66450404013575400\) \(-7817823581793132234600\) \([2]\) \(35831808\) \(3.0565\)  
324870.k3 324870k1 \([1, 1, 0, -338223, 60027633]\) \(36370300595789161/7754268658500\) \(912281953403866500\) \([2]\) \(5971968\) \(2.1606\) \(\Gamma_0(N)\)-optimal
324870.k4 324870k2 \([1, 1, 0, 738307, 365546847]\) \(378307987602100919/708763802906250\) \(-83385352648117406250\) \([2]\) \(11943936\) \(2.5072\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 324870.2.a.k

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.