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Properties

Label 19573a
Number of curves $3$
Conductor $19573$
CM no
Rank $2$
Graph

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

Elliptic curves in class 19573a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
19573.c3 19573a1 \([0, 1, 1, -1763, -28865]\) \(4096000/37\) \(5477327893\) \([]\) \(7920\) \(0.69122\) \(\Gamma_0(N)\)-optimal
19573.c2 19573a2 \([0, 1, 1, -12343, 507012]\) \(1404928000/50653\) \(7498461885517\) \([]\) \(23760\) \(1.2405\)  
19573.c1 19573a3 \([0, 1, 1, -990993, 379381573]\) \(727057727488000/37\) \(5477327893\) \([]\) \(71280\) \(1.7898\)  

Rank

sage: E.rank()
 

The elliptic curves in class 19573a have rank \(2\).

Complex multiplication

The elliptic curves in class 19573a do not have complex multiplication.

Modular form 19573.2.a.a

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