Properties

Label 54978.bo
Number of curves $4$
Conductor $54978$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bo1")
 
E.isogeny_class()
 

Elliptic curves in class 54978.bo

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
54978.bo1 54978br3 \([1, 1, 1, -813133833, 8924310531639]\) \(505384091400037554067434625/815656731648\) \(95961198821655552\) \([2]\) \(12441600\) \(3.4139\)  
54978.bo2 54978br4 \([1, 1, 1, -813125993, 8924491237367]\) \(-505369473241574671219626625/20303219722982711328\) \(-2388653497189193005027872\) \([2]\) \(24883200\) \(3.7605\)  
54978.bo3 54978br1 \([1, 1, 1, -10066953, 12165719607]\) \(959024269496848362625/11151660319506432\) \(1311981684929612218368\) \([2]\) \(4147200\) \(2.8646\) \(\Gamma_0(N)\)-optimal
54978.bo4 54978br2 \([1, 1, 1, -2038793, 31044740663]\) \(-7966267523043306625/3534510366354604032\) \(-415831610091252809760768\) \([2]\) \(8294400\) \(3.2112\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 54978.2.a.bo

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.