Properties

Label 10710.d
Number of curves $2$
Conductor $10710$
CM no
Rank $1$
Graph

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

Elliptic curves in class 10710.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10710.d1 10710b2 \([1, -1, 0, -197520, 33837596]\) \(43297905398453523/9912700\) \(195111674100\) \([2]\) \(59904\) \(1.5473\)  
10710.d2 10710b1 \([1, -1, 0, -12300, 535040]\) \(-10456049121363/160002640\) \(-3149331963120\) \([2]\) \(29952\) \(1.2007\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 10710.d have rank \(1\).

Complex multiplication

The elliptic curves in class 10710.d do not have complex multiplication.

Modular form 10710.2.a.d

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

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.