Properties

Label 435344p
Number of curves $2$
Conductor $435344$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("p1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 435344p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
435344.p1 435344p1 \([0, -1, 0, -31997, -5187599]\) \(-2932006912/7750379\) \(-9576857372316416\) \([]\) \(2322432\) \(1.7541\) \(\Gamma_0(N)\)-optimal*
435344.p2 435344p2 \([0, -1, 0, 278963, 117330641]\) \(1942951190528/5944921619\) \(-7345920300770245376\) \([]\) \(6967296\) \(2.3034\) \(\Gamma_0(N)\)-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 435344p1.

Rank

sage: E.rank()
 

The elliptic curves in class 435344p have rank \(1\).

Complex multiplication

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

Modular form 435344.2.a.p

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.