Properties

Label 592.a
Number of curves $3$
Conductor $592$
CM no
Rank $1$
Graph

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

Elliptic curves in class 592.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
592.a1 592e3 \([0, -1, 0, -29973, 2007325]\) \(727057727488000/37\) \(151552\) \([]\) \(432\) \(0.91523\)  
592.a2 592e2 \([0, -1, 0, -373, 2813]\) \(1404928000/50653\) \(207474688\) \([]\) \(144\) \(0.36592\)  
592.a3 592e1 \([0, -1, 0, -53, -131]\) \(4096000/37\) \(151552\) \([]\) \(48\) \(-0.18338\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 592.a have rank \(1\).

Complex multiplication

The elliptic curves in class 592.a do not have complex multiplication.

Modular form 592.2.a.a

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