Properties

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

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Show commands: SageMath
E = EllipticCurve("bh1")
 
E.isogeny_class()
 

Elliptic curves in class 54978bh

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
54978.q3 54978bh1 \([1, 0, 1, -405697, 89459444]\) \(62768149033310713/6915442583808\) \(813594904542427392\) \([2]\) \(1105920\) \(2.1701\) \(\Gamma_0(N)\)-optimal
54978.q2 54978bh2 \([1, 0, 1, -1538577, -638755820]\) \(3423676911662954233/483711578981136\) \(56908183555551669264\) \([2, 2]\) \(2211840\) \(2.5166\)  
54978.q4 54978bh3 \([1, 0, 1, 2509803, -3432138020]\) \(14861225463775641287/51859390496937804\) \(-6101205432574235702796\) \([2]\) \(4423680\) \(2.8632\)  
54978.q1 54978bh4 \([1, 0, 1, -23713037, -44446618996]\) \(12534210458299016895673/315581882565708\) \(37127892901972980492\) \([2]\) \(4423680\) \(2.8632\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 54978.2.a.bh

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.