Properties

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

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

Elliptic curves in class 198.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
198.d1 198c4 \([1, -1, 1, -1325, 4969]\) \(13060888875/7086244\) \(139478540652\) \([2]\) \(192\) \(0.82936\)  
198.d2 198c2 \([1, -1, 1, -1025, 12881]\) \(4406910829875/7744\) \(209088\) \([6]\) \(64\) \(0.28006\)  
198.d3 198c3 \([1, -1, 1, -785, -8207]\) \(2714704875/21296\) \(419169168\) \([2]\) \(96\) \(0.48279\)  
198.d4 198c1 \([1, -1, 1, -65, 209]\) \(1108717875/45056\) \(1216512\) \([6]\) \(32\) \(-0.066515\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 198.2.a.d

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.