Properties

Label 195.a
Number of curves $8$
Conductor $195$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 195.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
195.a1 195a7 \([1, 0, 0, -130000, -18051943]\) \(242970740812818720001/24375\) \(24375\) \([2]\) \(384\) \(1.1894\)  
195.a2 195a5 \([1, 0, 0, -8125, -282568]\) \(59319456301170001/594140625\) \(594140625\) \([2, 2]\) \(192\) \(0.84284\)  
195.a3 195a8 \([1, 0, 0, -7930, -296725]\) \(-55150149867714721/5950927734375\) \(-5950927734375\) \([2]\) \(384\) \(1.1894\)  
195.a4 195a3 \([1, 0, 0, -520, -4225]\) \(15551989015681/1445900625\) \(1445900625\) \([2, 4]\) \(96\) \(0.49627\)  
195.a5 195a2 \([1, 0, 0, -115, 392]\) \(168288035761/27720225\) \(27720225\) \([2, 4]\) \(48\) \(0.14970\)  
195.a6 195a1 \([1, 0, 0, -110, 435]\) \(147281603041/5265\) \(5265\) \([4]\) \(24\) \(-0.19688\) \(\Gamma_0(N)\)-optimal
195.a7 195a4 \([1, 0, 0, 210, 2277]\) \(1023887723039/2798036865\) \(-2798036865\) \([4]\) \(96\) \(0.49627\)  
195.a8 195a6 \([1, 0, 0, 605, -19750]\) \(24487529386319/183539412225\) \(-183539412225\) \([4]\) \(192\) \(0.84284\)  

Rank

sage: E.rank()
 

The elliptic curves in class 195.a have rank \(0\).

Complex multiplication

The elliptic curves in class 195.a do not have complex multiplication.

Modular form 195.2.a.a

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

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 16 & 16 & 8 \\ 2 & 1 & 2 & 2 & 4 & 8 & 8 & 4 \\ 4 & 2 & 1 & 4 & 8 & 16 & 16 & 8 \\ 4 & 2 & 4 & 1 & 2 & 4 & 4 & 2 \\ 8 & 4 & 8 & 2 & 1 & 2 & 2 & 4 \\ 16 & 8 & 16 & 4 & 2 & 1 & 4 & 8 \\ 16 & 8 & 16 & 4 & 2 & 4 & 1 & 8 \\ 8 & 4 & 8 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.