Properties

Label 198.e
Number of curves $4$
Conductor $198$
CM no
Rank $0$
Graph

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

Elliptic curves in class 198.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
198.e1 198b3 \([1, -1, 1, -725, 7661]\) \(57736239625/255552\) \(186297408\) \([6]\) \(96\) \(0.43923\)  
198.e2 198b4 \([1, -1, 1, -365, 15005]\) \(-7357983625/127552392\) \(-92985693768\) \([6]\) \(192\) \(0.78580\)  
198.e3 198b1 \([1, -1, 1, -50, -115]\) \(18609625/1188\) \(866052\) \([2]\) \(32\) \(-0.11008\) \(\Gamma_0(N)\)-optimal
198.e4 198b2 \([1, -1, 1, 40, -547]\) \(9938375/176418\) \(-128608722\) \([2]\) \(64\) \(0.23650\)  

Rank

sage: E.rank()
 

The elliptic curves in class 198.e have rank \(0\).

Complex multiplication

The elliptic curves in class 198.e do not have complex multiplication.

Modular form 198.2.a.e

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.