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Properties

Label 710.d
Number of curves $2$
Conductor $710$
CM no
Rank $0$
Graph

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

Elliptic curves in class 710.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
710.d1 710d2 \([1, 1, 1, -181355, -29801973]\) \(659648323242974383921/90211467550\) \(90211467550\) \([]\) \(3600\) \(1.5145\)  
710.d2 710d1 \([1, 1, 1, -1105, 11727]\) \(149222774347921/22187500000\) \(22187500000\) \([5]\) \(720\) \(0.70973\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 710.d have rank \(0\).

Complex multiplication

The elliptic curves in class 710.d do not have complex multiplication.

Modular form 710.2.a.d

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

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.