Properties

Label 130130.f
Number of curves $4$
Conductor $130130$
CM no
Rank $2$
Graph

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

Elliptic curves in class 130130.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
130130.f1 130130f4 \([1, 0, 1, -594884, -175688004]\) \(4823468134087681/30382271150\) \(146649419827260350\) \([2]\) \(2654208\) \(2.1313\)  
130130.f2 130130f2 \([1, 0, 1, -45634, 3587196]\) \(2177286259681/105875000\) \(511038402875000\) \([2]\) \(884736\) \(1.5820\)  
130130.f3 130130f3 \([1, 0, 1, -15214, -5960628]\) \(-80677568161/3131816380\) \(-15116679489331420\) \([2]\) \(1327104\) \(1.7847\)  
130130.f4 130130f1 \([1, 0, 1, 1686, 218012]\) \(109902239/4312000\) \(-20813200408000\) \([2]\) \(442368\) \(1.2354\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 130130.f have rank \(2\).

Complex multiplication

The elliptic curves in class 130130.f do not have complex multiplication.

Modular form 130130.2.a.f

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

\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.