Properties

Label 257600.fa
Number of curves $2$
Conductor $257600$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("fa1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 257600.fa

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
257600.fa1 257600fa2 \([0, -1, 0, -968833, 343953537]\) \(24553362849625/1755162752\) \(7189146632192000000\) \([2]\) \(6193152\) \(2.3652\)  
257600.fa2 257600fa1 \([0, -1, 0, 55167, 23441537]\) \(4533086375/60669952\) \(-248504123392000000\) \([2]\) \(3096576\) \(2.0186\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 257600.fa have rank \(2\).

Complex multiplication

The elliptic curves in class 257600.fa do not have complex multiplication.

Modular form 257600.2.a.fa

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

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.