Properties

Label 450840.bl
Number of curves $4$
Conductor $450840$
CM no
Rank $0$
Graph

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

Elliptic curves in class 450840.bl

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.bl1 450840bl3 \([0, 1, 0, -2404576, 1434377120]\) \(31103978031362/195\) \(9639579555840\) \([2]\) \(7077888\) \(2.0970\) \(\Gamma_0(N)\)-optimal*
450840.bl2 450840bl4 \([0, 1, 0, -208176, 3526560]\) \(20183398562/11567205\) \(571810219672872960\) \([2]\) \(7077888\) \(2.0970\)  
450840.bl3 450840bl2 \([0, 1, 0, -150376, 22346240]\) \(15214885924/38025\) \(939859006694400\) \([2, 2]\) \(3538944\) \(1.7504\) \(\Gamma_0(N)\)-optimal*
450840.bl4 450840bl1 \([0, 1, 0, -5876, 613440]\) \(-3631696/24375\) \(-150618430560000\) \([2]\) \(1769472\) \(1.4038\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 450840.bl1.

Rank

sage: E.rank()
 

The elliptic curves in class 450840.bl have rank \(0\).

Complex multiplication

The elliptic curves in class 450840.bl do not have complex multiplication.

Modular form 450840.2.a.bl

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{5} - 4q^{7} + q^{9} - 4q^{11} + q^{13} - q^{15} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.