Properties

Label 51842d
Number of curves $2$
Conductor $51842$
CM no
Rank $1$
Graph

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

Elliptic curves in class 51842d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51842.h2 51842d1 \([1, -1, 0, -264070, -60576972]\) \(-116930169/23552\) \(-410188092430441472\) \([2]\) \(760320\) \(2.1022\) \(\Gamma_0(N)\)-optimal
51842.h1 51842d2 \([1, -1, 0, -4411430, -3565096172]\) \(545138290809/16928\) \(294822691434379808\) \([2]\) \(1520640\) \(2.4488\)  

Rank

sage: E.rank()
 

The elliptic curves in class 51842d have rank \(1\).

Complex multiplication

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

Modular form 51842.2.a.d

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