Properties

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

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

Elliptic curves in class 19a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
19.a2 19a1 \([0, 1, 1, -9, -15]\) \(-89915392/6859\) \(-6859\) \([3]\) \(1\) \(-0.51587\) \(\Gamma_0(N)\)-optimal
19.a1 19a2 \([0, 1, 1, -769, -8470]\) \(-50357871050752/19\) \(-19\) \([]\) \(3\) \(0.033439\)  
19.a3 19a3 \([0, 1, 1, 1, 0]\) \(32768/19\) \(-19\) \([3]\) \(3\) \(-1.0652\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 19a do not have complex multiplication.

Modular form 19.2.a.a

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.