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Properties

Label 195195.j
Number of curves $4$
Conductor $195195$
CM no
Rank $1$
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Show commands: SageMath
E = EllipticCurve("j1")
 
E.isogeny_class()
 

Elliptic curves in class 195195.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
195195.j1 195195i3 \([1, 0, 0, -97177961, 368714524710]\) \(21026497979043461623321/161783881875\) \(780899897089186875\) \([2]\) \(14745600\) \(3.0258\)  
195195.j2 195195i2 \([1, 0, 0, -6077666, 5752729371]\) \(5143681768032498601/14238434358225\) \(68726203106189654025\) \([2, 2]\) \(7372800\) \(2.6792\)  
195195.j3 195195i4 \([1, 0, 0, -3682091, 10334506116]\) \(-1143792273008057401/8897444448004035\) \(-42946264938625908174315\) \([2]\) \(14745600\) \(3.0258\)  
195195.j4 195195i1 \([1, 0, 0, -533621, 10207560]\) \(3481467828171481/2005331497785\) \(9679352121492118065\) \([2]\) \(3686400\) \(2.3326\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 195195.2.a.j

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.