Properties

Label 232050ha
Number of curves $4$
Conductor $232050$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ha1")
 
E.isogeny_class()
 

Elliptic curves in class 232050ha

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
232050.ha3 232050ha1 \([1, 0, 0, -254338, 49348292]\) \(116449478628435289/1996001280\) \(31187520000000\) \([2]\) \(1769472\) \(1.7186\) \(\Gamma_0(N)\)-optimal
232050.ha2 232050ha2 \([1, 0, 0, -262338, 46076292]\) \(127787213284071769/15197834433600\) \(237466163025000000\) \([2, 2]\) \(3538944\) \(2.0651\)  
232050.ha4 232050ha3 \([1, 0, 0, 374662, 235265292]\) \(372239584720800551/1745320379985000\) \(-27270630937265625000\) \([2]\) \(7077888\) \(2.4117\)  
232050.ha1 232050ha4 \([1, 0, 0, -1027338, -352488708]\) \(7674388308884766169/1007648705929320\) \(15744511030145625000\) \([2]\) \(7077888\) \(2.4117\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 232050.2.a.ha

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.