Properties

Label 19.a
Number of curves 3
Conductor 19
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("19.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 19.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
19.a1 19a2 [0, 1, 1, -769, -8470] [] 3  
19.a2 19a1 [0, 1, 1, -9, -15] [3] 1 \(\Gamma_0(N)\)-optimal
19.a3 19a3 [0, 1, 1, 1, 0] [3] 3  

Rank

sage: E.rank()
 

The elliptic curves in class 19.a have rank \(0\).

Modular form 19.2.a.a

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

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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.