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Properties

Label 257754.q
Number of curves $2$
Conductor $257754$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("q1")
 
E.isogeny_class()
 

Elliptic curves in class 257754.q

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
257754.q1 257754q2 \([1, 1, 0, -4443936, 4091116032]\) \(-26885619637842619882657/4488366981249252864\) \(-1620300480230980283904\) \([]\) \(17635968\) \(2.7967\)  
257754.q2 257754q1 \([1, 1, 0, 371424, -22514688]\) \(15697286016456868703/9699053927399424\) \(-3501358467791192064\) \([]\) \(5878656\) \(2.2474\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 257754.q have rank \(1\).

Complex multiplication

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

Modular form 257754.2.a.q

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + 3 q^{5} + q^{6} + q^{7} - q^{8} + q^{9} - 3 q^{10} - q^{12} - 5 q^{13} - q^{14} - 3 q^{15} + q^{16} - q^{17} - q^{18} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.