Properties

Label 98.a
Number of curves $6$
Conductor $98$
CM no
Rank $0$
Graph

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

Elliptic curves in class 98.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
98.a1 98a6 \([1, 1, 0, -133795, 18781197]\) \(2251439055699625/25088\) \(2951578112\) \([2]\) \(288\) \(1.3861\)  
98.a2 98a5 \([1, 1, 0, -8355, 291341]\) \(-548347731625/1835008\) \(-215886856192\) \([2]\) \(144\) \(1.0395\)  
98.a3 98a4 \([1, 1, 0, -1740, 22184]\) \(4956477625/941192\) \(110730297608\) \([2]\) \(96\) \(0.83675\)  
98.a4 98a2 \([1, 1, 0, -515, -4717]\) \(128787625/98\) \(11529602\) \([2]\) \(32\) \(0.28744\)  
98.a5 98a1 \([1, 1, 0, -25, -111]\) \(-15625/28\) \(-3294172\) \([2]\) \(16\) \(-0.059130\) \(\Gamma_0(N)\)-optimal
98.a6 98a3 \([1, 1, 0, 220, 2192]\) \(9938375/21952\) \(-2582630848\) \([2]\) \(48\) \(0.49018\)  

Rank

sage: E.rank()
 

The elliptic curves in class 98.a have rank \(0\).

Complex multiplication

The elliptic curves in class 98.a do not have complex multiplication.

Modular form 98.2.a.a

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.