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Properties

Label 164730.f
Number of curves $2$
Conductor $164730$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 164730.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
164730.f1 164730dc1 \([1, 1, 0, -72499413157873, 237600840225377739733]\) \(1745957458089824793658821537153909697081/8482844302577646464705495040000\) \(204755239669724819399434951207157760000\) \([2]\) \(19646668800\) \(6.5516\) \(\Gamma_0(N)\)-optimal
164730.f2 164730dc2 \([1, 1, 0, -71287259301873, 245929307666474853333]\) \(-1659838900070008272993828621295780801081/121902690479959282132916661701836800\) \(-2942434602745660289673743093077743186739200\) \([2]\) \(39293337600\) \(6.8981\)  

Rank

sage: E.rank()
 

The elliptic curves in class 164730.f have rank \(1\).

Complex multiplication

The elliptic curves in class 164730.f do not have complex multiplication.

Modular form 164730.2.a.f

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.