Properties

Label 123200.ha
Number of curves 4
Conductor 123200
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("123200.ha1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 123200.ha

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
123200.ha1 123200fs4 [0, -1, 0, -41785633, 101033299137] [2] 15925248  
123200.ha2 123200fs2 [0, -1, 0, -5721633, -5219084863] [2] 5308416  
123200.ha3 123200fs1 [0, -1, 0, -89633, -200972863] [2] 2654208 \(\Gamma_0(N)\)-optimal
123200.ha4 123200fs3 [0, -1, 0, 806367, 5414259137] [2] 7962624  

Rank

sage: E.rank()
 

The elliptic curves in class 123200.ha have rank \(1\).

Modular form 123200.2.a.ha

sage: E.q_eigenform(10)
 
\( q + 2q^{3} + q^{7} + q^{9} - q^{11} - 4q^{13} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.