Properties

Label 504.f
Number of curves $2$
Conductor $504$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 504.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
504.f1 504d2 \([0, 0, 0, -999, -12150]\) \(21882096/7\) \(35271936\) \([2]\) \(192\) \(0.42301\)  
504.f2 504d1 \([0, 0, 0, -54, -243]\) \(-55296/49\) \(-15431472\) \([2]\) \(96\) \(0.076434\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 504.f have rank \(0\).

Complex multiplication

The elliptic curves in class 504.f do not have complex multiplication.

Modular form 504.2.a.f

sage: E.q_eigenform(10)
 
\(q + 2 q^{5} - q^{7} + 2 q^{11} + 2 q^{13} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.