Properties

Label 51842p
Number of curves $2$
Conductor $51842$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 51842p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51842.m2 51842p1 \([1, 0, 0, -104224, 11897724]\) \(7189057/644\) \(11216080652394884\) \([2]\) \(608256\) \(1.8194\) \(\Gamma_0(N)\)-optimal
51842.m1 51842p2 \([1, 0, 0, -363434, -70893950]\) \(304821217/51842\) \(902894492517788162\) \([2]\) \(1216512\) \(2.1660\)  

Rank

sage: E.rank()
 

The elliptic curves in class 51842p have rank \(0\).

Complex multiplication

The elliptic curves in class 51842p do not have complex multiplication.

Modular form 51842.2.a.p

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