Properties

Label 194271d
Number of curves $4$
Conductor $194271$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 194271d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
194271.i3 194271d1 \([1, 0, 0, -129952, -14661025]\) \(408023180713/80247321\) \(47732977978573041\) \([2]\) \(1290240\) \(1.9169\) \(\Gamma_0(N)\)-optimal
194271.i2 194271d2 \([1, 0, 0, -638757, 183264120]\) \(48455467135993/3635004681\) \(2162185556202965601\) \([2, 2]\) \(2580480\) \(2.2635\)  
194271.i1 194271d3 \([1, 0, 0, -10028522, 12222820803]\) \(187519537050946633/1186707753\) \(705881446695907713\) \([2]\) \(5160960\) \(2.6101\)  
194271.i4 194271d4 \([1, 0, 0, 610128, 812951937]\) \(42227808999767/504359959257\) \(-300005065944673432497\) \([2]\) \(5160960\) \(2.6101\)  

Rank

sage: E.rank()
 

The elliptic curves in class 194271d have rank \(1\).

Complex multiplication

The elliptic curves in class 194271d do not have complex multiplication.

Modular form 194271.2.a.d

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.