Properties

Label 8007.a
Number of curves $2$
Conductor $8007$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 8007.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8007.a1 8007a2 \([1, 1, 0, -840, -9729]\) \(65670324063625/3675213\) \(3675213\) \([2]\) \(1920\) \(0.32470\)  
8007.a2 8007a1 \([1, 1, 0, -55, -152]\) \(18927429625/3771297\) \(3771297\) \([2]\) \(960\) \(-0.021869\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 8007.a have rank \(1\).

Complex multiplication

The elliptic curves in class 8007.a do not have complex multiplication.

Modular form 8007.2.a.a

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} - q^{4} - q^{6} + 2 q^{7} - 3 q^{8} + q^{9} + 4 q^{11} + q^{12} + 2 q^{13} + 2 q^{14} - q^{16} + q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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 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.