Properties

Label 1002.d
Number of curves $2$
Conductor $1002$
CM no
Rank $1$
Graph

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

Elliptic curves in class 1002.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1002.d1 1002d2 \([1, 0, 1, -125, -544]\) \(213525509833/669336\) \(669336\) \([2]\) \(288\) \(-0.014852\)  
1002.d2 1002d1 \([1, 0, 1, -5, -16]\) \(-10218313/96192\) \(-96192\) \([2]\) \(144\) \(-0.36143\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 1002.d have rank \(1\).

Complex multiplication

The elliptic curves in class 1002.d do not have complex multiplication.

Modular form 1002.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} + 2 q^{5} - q^{6} - 4 q^{7} - q^{8} + q^{9} - 2 q^{10} - 4 q^{11} + q^{12} + 4 q^{14} + 2 q^{15} + q^{16} - 4 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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 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.