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Properties

Label 257754j
Number of curves $2$
Conductor $257754$
CM no
Rank $2$
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Show commands: SageMath
E = EllipticCurve("j1")
 
E.isogeny_class()
 

Elliptic curves in class 257754j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
257754.j1 257754j1 \([1, 1, 0, -16765, 828541]\) \(75982583888875/2718912\) \(18649017408\) \([2]\) \(460800\) \(1.0598\) \(\Gamma_0(N)\)-optimal
257754.j2 257754j2 \([1, 1, 0, -16005, 908037]\) \(-66110789312875/14438442312\) \(-99033275818008\) \([2]\) \(921600\) \(1.4063\)  

Rank

sage: E.rank()
 

The elliptic curves in class 257754j have rank \(2\).

Complex multiplication

The elliptic curves in class 257754j do not have complex multiplication.

Modular form 257754.2.a.j

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} - q^{7} - q^{8} + q^{9} - 2 q^{11} - q^{12} - 6 q^{13} + q^{14} + q^{16} + q^{17} - q^{18} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.