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Properties

Label 100002.a
Number of curves $2$
Conductor $100002$
CM no
Rank $1$
Graph

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

Elliptic curves in class 100002.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
100002.a1 100002a2 \([1, 1, 0, -448, -560]\) \(9978645018889/5714514288\) \(5714514288\) \([2]\) \(150272\) \(0.56189\)  
100002.a2 100002a1 \([1, 1, 0, 112, 0]\) \(153216258551/89601792\) \(-89601792\) \([2]\) \(75136\) \(0.21532\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 100002.a have rank \(1\).

Complex multiplication

The elliptic curves in class 100002.a do not have complex multiplication.

Modular form 100002.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + 4 q^{5} + q^{6} - q^{7} - q^{8} + q^{9} - 4 q^{10} - 6 q^{11} - q^{12} + 4 q^{13} + q^{14} - 4 q^{15} + q^{16} - 4 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.