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Properties

Label 197106ce
Number of curves $2$
Conductor $197106$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("ce1")
 
E.isogeny_class()
 

Elliptic curves in class 197106ce

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
197106.g2 197106ce1 \([1, 1, 0, 257747, 569181469]\) \(40251338884511/2997011332224\) \(-140997038491461769344\) \([]\) \(8446032\) \(2.5458\) \(\Gamma_0(N)\)-optimal
197106.g1 197106ce2 \([1, 1, 0, -1326493063, 18594850025839]\) \(-5486773802537974663600129/2635437714\) \(-123986489075756034\) \([]\) \(59122224\) \(3.5188\)  

Rank

sage: E.rank()
 

The elliptic curves in class 197106ce have rank \(0\).

Complex multiplication

The elliptic curves in class 197106ce do not have complex multiplication.

Modular form 197106.2.a.ce

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

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.