Properties

Label 414736.bz
Number of curves $2$
Conductor $414736$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("bz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 414736.bz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
414736.bz1 414736bz2 \([0, -1, 0, -251131288, -1433983312656]\) \(24553362849625/1755162752\) \(125208149764842438707904512\) \([2]\) \(136249344\) \(3.7546\) \(\Gamma_0(N)\)-optimal*
414736.bz2 414736bz1 \([0, -1, 0, 14299752, -97909629712]\) \(4533086375/60669952\) \(-4328015978908947381747712\) \([2]\) \(68124672\) \(3.4080\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 414736.bz1.

Rank

sage: E.rank()
 

The elliptic curves in class 414736.bz have rank \(0\).

Complex multiplication

The elliptic curves in class 414736.bz do not have complex multiplication.

Modular form 414736.2.a.bz

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.