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Properties

Label 51842.d
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 51842.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51842.d1 51842i2 \([1, 0, 1, -15695706, 22405989260]\) \(24553362849625/1755162752\) \(30568395938682236012672\) \([2]\) \(5677056\) \(3.0615\)  
51842.d2 51842i1 \([1, 0, 1, 893734, 1529837964]\) \(4533086375/60669952\) \(-1056644526100817231872\) \([2]\) \(2838528\) \(2.7149\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 51842.2.a.d

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.