Properties

Label 75810.l
Number of curves $8$
Conductor $75810$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("75810.l1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 75810.l

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
75810.l1 75810l8 [1, 1, 0, -2328818, -861559362] [2] 3981312  
75810.l2 75810l5 [1, 1, 0, -2079728, -1155271128] [2] 1327104  
75810.l3 75810l6 [1, 1, 0, -975068, 360335388] [2, 2] 1990656  
75810.l4 75810l3 [1, 1, 0, -967848, 366083952] [2] 995328  
75810.l5 75810l2 [1, 1, 0, -130328, -17991168] [2, 2] 663552  
75810.l6 75810l4 [1, 1, 0, -29248, -45060392] [2] 1327104  
75810.l7 75810l1 [1, 1, 0, -14808, 237888] [2] 331776 \(\Gamma_0(N)\)-optimal
75810.l8 75810l7 [1, 1, 0, 263162, 1214466442] [2] 3981312  

Rank

sage: E.rank()
 

The elliptic curves in class 75810.l have rank \(1\).

Modular form 75810.2.a.l

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + q^{7} - q^{8} + q^{9} + q^{10} - q^{12} - 2q^{13} - q^{14} + q^{15} + q^{16} - 6q^{17} - q^{18} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.