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Properties

Label 195195.e
Number of curves $2$
Conductor $195195$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 195195.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
195195.e1 195195o2 \([1, 1, 1, -725605, -237958648]\) \(3984138055477/4764375\) \(50523811700236875\) \([2]\) \(3035136\) \(2.1161\)  
195195.e2 195195o1 \([1, 1, 1, -33550, -5704990]\) \(-393832837/1091475\) \(-11574545953145175\) \([2]\) \(1517568\) \(1.7696\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 195195.e have rank \(1\).

Complex multiplication

The elliptic curves in class 195195.e do not have complex multiplication.

Modular form 195195.2.a.e

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