Properties

Label 56784bl
Number of curves $4$
Conductor $56784$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bl1")
 
E.isogeny_class()
 

Elliptic curves in class 56784bl

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
56784.bd3 56784bl1 \([0, -1, 0, -152832, 22413312]\) \(19968681097/628992\) \(12435555313778688\) \([2]\) \(387072\) \(1.8631\) \(\Gamma_0(N)\)-optimal
56784.bd2 56784bl2 \([0, -1, 0, -369152, -54942720]\) \(281397674377/96589584\) \(1909634962872139776\) \([2, 2]\) \(774144\) \(2.2097\)  
56784.bd4 56784bl3 \([0, -1, 0, 1091008, -383186688]\) \(7264187703863/7406095788\) \(-146423028958742495232\) \([2]\) \(1548288\) \(2.5563\)  
56784.bd1 56784bl4 \([0, -1, 0, -5290432, -4680945920]\) \(828279937799497/193444524\) \(3824516175642279936\) \([2]\) \(1548288\) \(2.5563\)  

Rank

sage: E.rank()
 

The elliptic curves in class 56784bl have rank \(0\).

Complex multiplication

The elliptic curves in class 56784bl do not have complex multiplication.

Modular form 56784.2.a.bl

sage: E.q_eigenform(10)
 
\(q - q^{3} + 2 q^{5} - q^{7} + q^{9} - 4 q^{11} - 2 q^{15} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 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.