Properties

Label 4998r
Number of curves $2$
Conductor $4998$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 4998r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4998.t2 4998r1 \([1, 0, 1, -180, 23986]\) \(-1865409391/724451328\) \(-248486805504\) \([2]\) \(8064\) \(0.86570\) \(\Gamma_0(N)\)-optimal
4998.t1 4998r2 \([1, 0, 1, -13620, 604594]\) \(814544990575471/9268826496\) \(3179207488128\) \([2]\) \(16128\) \(1.2123\)  

Rank

sage: E.rank()
 

The elliptic curves in class 4998r have rank \(1\).

Complex multiplication

The elliptic curves in class 4998r do not have complex multiplication.

Modular form 4998.2.a.r

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} + 2 q^{5} - q^{6} - q^{8} + q^{9} - 2 q^{10} - 4 q^{11} + q^{12} + 2 q^{13} + 2 q^{15} + q^{16} - q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.