Properties

Label 1785h
Number of curves 4
Conductor 1785
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("1785.b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1785h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1785.b3 1785h1 [1, 1, 1, -335, -2500] [2] 384 \(\Gamma_0(N)\)-optimal
1785.b2 1785h2 [1, 1, 1, -340, -2428] [2, 2] 768  
1785.b1 1785h3 [1, 1, 1, -1015, 9182] [4] 1536  
1785.b4 1785h4 [1, 1, 1, 255, -9330] [2] 1536  

Rank

sage: E.rank()
 

The elliptic curves in class 1785h have rank \(0\).

Modular form 1785.2.a.b

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + q^{7} + 3q^{8} + q^{9} - q^{10} + 4q^{11} + q^{12} + 2q^{13} - q^{14} - q^{15} - q^{16} + q^{17} - q^{18} + 8q^{19} + O(q^{20}) \)

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 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.