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Properties

Label 232050ge
Number of curves $2$
Conductor $232050$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ge1")
 
E.isogeny_class()
 

Elliptic curves in class 232050ge

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
232050.ge2 232050ge1 \([1, 0, 0, 4812, -721008]\) \(788632918919/14845259520\) \(-231957180000000\) \([2]\) \(1032192\) \(1.4369\) \(\Gamma_0(N)\)-optimal
232050.ge1 232050ge2 \([1, 0, 0, -97188, -11023008]\) \(6497434355239801/405606692400\) \(6337604568750000\) \([2]\) \(2064384\) \(1.7835\)  

Rank

sage: E.rank()
 

The elliptic curves in class 232050ge have rank \(1\).

Complex multiplication

The elliptic curves in class 232050ge do not have complex multiplication.

Modular form 232050.2.a.ge

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{6} - q^{7} + q^{8} + q^{9} - 2 q^{11} + q^{12} + q^{13} - q^{14} + q^{16} + q^{17} + q^{18} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.