Properties

Label 105710.b
Number of curves $2$
Conductor $105710$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 105710.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
105710.b1 105710a1 \([1, 1, 0, -85068, -9676592]\) \(-76711450249/851840\) \(-756011135623040\) \([]\) \(819000\) \(1.6698\) \(\Gamma_0(N)\)-optimal
105710.b2 105710a2 \([1, 1, 0, 284917, -49856963]\) \(2882081488391/2883584000\) \(-2559191414472704000\) \([]\) \(2457000\) \(2.2191\)  

Rank

sage: E.rank()
 

The elliptic curves in class 105710.b have rank \(0\).

Complex multiplication

The elliptic curves in class 105710.b do not have complex multiplication.

Modular form 105710.2.a.b

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