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Properties

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

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Show commands: SageMath
E = EllipticCurve("fb1")
 
E.isogeny_class()
 

Elliptic curves in class 324870.fb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
324870.fb1 324870fb4 \([1, 0, 0, -3754381, -2795530375]\) \(49745123032831462081/97939634471640\) \(11522500055953974360\) \([2]\) \(13271040\) \(2.5457\)  
324870.fb2 324870fb3 \([1, 0, 0, -3146781, 2137150665]\) \(29291056630578924481/175463302795560\) \(20643082110594838440\) \([2]\) \(13271040\) \(2.5457\)  
324870.fb3 324870fb2 \([1, 0, 0, -314581, -11356255]\) \(29263955267177281/16463793153600\) \(1936948800727886400\) \([2, 2]\) \(6635520\) \(2.1991\)  
324870.fb4 324870fb1 \([1, 0, 0, 77419, -1399455]\) \(436192097814719/259683840000\) \(-30551544092160000\) \([2]\) \(3317760\) \(1.8525\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 324870.2.a.fb

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

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.