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Properties

Label 1710.d
Number of curves $2$
Conductor $1710$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 1710.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1710.d1 1710e2 \([1, -1, 0, -25020, 1529550]\) \(-2376117230685121/342950\) \(-250010550\) \([3]\) \(2160\) \(1.0204\)  
1710.d2 1710e1 \([1, -1, 0, -270, 2700]\) \(-2992209121/2375000\) \(-1731375000\) \([]\) \(720\) \(0.47114\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 1710.d have rank \(1\).

Complex multiplication

The elliptic curves in class 1710.d do not have complex multiplication.

Modular form 1710.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{5} - q^{7} - q^{8} + q^{10} - q^{13} + q^{14} + q^{16} + 3 q^{17} + q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.