Properties

Label 1710d
Number of curves $4$
Conductor $1710$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("1710.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1710d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1710.f4 1710d1 [1, -1, 0, -90, -540] [2] 768 \(\Gamma_0(N)\)-optimal
1710.f3 1710d2 [1, -1, 0, -1710, -26784] [2, 2] 1536  
1710.f1 1710d3 [1, -1, 0, -27360, -1735074] [2] 3072  
1710.f2 1710d4 [1, -1, 0, -1980, -17550] [2] 3072  

Rank

sage: E.rank()
 

The elliptic curves in class 1710d have rank \(0\).

Modular form 1710.2.a.f

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

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.