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Properties

Label 54978w
Number of curves $2$
Conductor $54978$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("w1")
 
E.isogeny_class()
 

Elliptic curves in class 54978w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
54978.n1 54978w1 \([1, 0, 1, -181963, 28785782]\) \(5663453071972249/231607799808\) \(27248426039611392\) \([2]\) \(1216512\) \(1.9193\) \(\Gamma_0(N)\)-optimal
54978.n2 54978w2 \([1, 0, 1, 84597, 106088182]\) \(569125098462311/41650447874112\) \(-4900133541941402688\) \([2]\) \(2433024\) \(2.2659\)  

Rank

sage: E.rank()
 

The elliptic curves in class 54978w have rank \(1\).

Complex multiplication

The elliptic curves in class 54978w do not have complex multiplication.

Modular form 54978.2.a.w

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - 4 q^{5} - q^{6} - q^{8} + q^{9} + 4 q^{10} - q^{11} + q^{12} + 6 q^{13} - 4 q^{15} + q^{16} - q^{17} - q^{18} + 6 q^{19} + O(q^{20})\) 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 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.