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Properties

Label 54978m
Number of curves $2$
Conductor $54978$
CM no
Rank $2$
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Show commands: SageMath
E = EllipticCurve("m1")
 
E.isogeny_class()
 

Elliptic curves in class 54978m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
54978.b2 54978m1 \([1, 1, 0, 2376, 352836]\) \(12600539783/461862324\) \(-54337640556276\) \([]\) \(221184\) \(1.3154\) \(\Gamma_0(N)\)-optimal
54978.b1 54978m2 \([1, 1, 0, -312939, 67272381]\) \(-28808239025774377/10677167424\) \(-1256158070266176\) \([]\) \(663552\) \(1.8647\)  

Rank

sage: E.rank()
 

The elliptic curves in class 54978m have rank \(2\).

Complex multiplication

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

Modular form 54978.2.a.m

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.