Properties

Label 435344o
Number of curves $3$
Conductor $435344$
CM no
Rank $2$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("o1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 435344o

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
435344.o3 435344o1 \([0, -1, 0, 43833643, -119695311971]\) \(471114356703100928/585612268875179\) \(-11577911582380580347129856\) \([]\) \(94058496\) \(3.4953\) \(\Gamma_0(N)\)-optimal*
435344.o2 435344o2 \([0, -1, 0, -431421397, 5369381586269]\) \(-449167881463536812032/369990050199923699\) \(-7314928862066456615512027136\) \([]\) \(282175488\) \(4.0446\) \(\Gamma_0(N)\)-optimal*
435344.o1 435344o3 \([0, -1, 0, -40099398837, 3090699934894909]\) \(-360675992659311050823073792/56219378022244619\) \(-1111491378430658671373398016\) \([]\) \(846526464\) \(4.5939\) \(\Gamma_0(N)\)-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 435344o1.

Rank

sage: E.rank()
 

The elliptic curves in class 435344o have rank \(2\).

Complex multiplication

The elliptic curves in class 435344o do not have complex multiplication.

Modular form 435344.2.a.o

sage: E.q_eigenform(10)
 
\(q - q^{3} - 3q^{5} + q^{7} - 2q^{9} - 3q^{11} + 3q^{15} - 6q^{17} - 7q^{19} + O(q^{20})\)  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 Cremona numbering.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.