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Properties

Label 1710i
Number of curves $2$
Conductor $1710$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("i1")
 
E.isogeny_class()
 

Elliptic curves in class 1710i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1710.k2 1710i1 \([1, -1, 0, -13104, 583308]\) \(-341370886042369/1817528220\) \(-1324978072380\) \([2]\) \(4480\) \(1.1708\) \(\Gamma_0(N)\)-optimal
1710.k1 1710i2 \([1, -1, 0, -209934, 37075590]\) \(1403607530712116449/39475350\) \(28777530150\) \([2]\) \(8960\) \(1.5174\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 1710.2.a.i

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