Properties

Label 128797.a
Number of curves $3$
Conductor $128797$
CM no
Rank $2$
Graph

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

Elliptic curves in class 128797.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
128797.a1 128797a3 \([0, 1, 1, -6521073, 6407364502]\) \(727057727488000/37\) \(1560679744717\) \([]\) \(1202688\) \(2.2608\)  
128797.a2 128797a2 \([0, 1, 1, -81223, 8600745]\) \(1404928000/50653\) \(2136570570517573\) \([]\) \(400896\) \(1.7115\)  
128797.a3 128797a1 \([0, 1, 1, -11603, -481184]\) \(4096000/37\) \(1560679744717\) \([]\) \(133632\) \(1.1622\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 128797.a have rank \(2\).

Complex multiplication

The elliptic curves in class 128797.a do not have complex multiplication.

Modular form 128797.2.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 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.