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Properties

Label 102.a
Number of curves $2$
Conductor $102$
CM no
Rank $1$
Graph

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

Elliptic curves in class 102.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
102.a1 102a1 \([1, 1, 0, -2, 0]\) \(1771561/612\) \(612\) \([2]\) \(8\) \(-0.76353\) \(\Gamma_0(N)\)-optimal
102.a2 102a2 \([1, 1, 0, 8, 10]\) \(46268279/46818\) \(-46818\) \([2]\) \(16\) \(-0.41695\)  

Rank

sage: E.rank()
 

The elliptic curves in class 102.a have rank \(1\).

Complex multiplication

The elliptic curves in class 102.a do not have complex multiplication.

Modular form 102.2.a.a

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