Properties

Label 453299q
Number of curves $4$
Conductor $453299$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("q1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 453299q

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
453299.q4 453299q1 \([1, -1, 0, 24078934, -51841081041]\) \(22062729659823/29354283343\) \(-2054223576913088525175847\) \([2]\) \(54190080\) \(3.3510\) \(\Gamma_0(N)\)-optimal*
453299.q3 453299q2 \([1, -1, 0, -149204911, -507334996008]\) \(5249244962308257/1448621666569\) \(101375078611919040499849201\) \([2, 2]\) \(108380160\) \(3.6976\) \(\Gamma_0(N)\)-optimal*
453299.q2 453299q3 \([1, -1, 0, -872216816, 9509994947767]\) \(1048626554636928177/48569076788309\) \(3398881910405717639146981661\) \([2]\) \(216760320\) \(4.0441\) \(\Gamma_0(N)\)-optimal*
453299.q1 453299q4 \([1, -1, 0, -2198734526, -39678354903811]\) \(16798320881842096017/2132227789307\) \(149214087258179015980526003\) \([2]\) \(216760320\) \(4.0441\)  
*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 453299q1.

Rank

sage: E.rank()
 

The elliptic curves in class 453299q have rank \(1\).

Complex multiplication

The elliptic curves in class 453299q do not have complex multiplication.

Modular form 453299.2.a.q

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{4} + 2q^{5} - 3q^{8} - 3q^{9} + 2q^{10} + q^{11} - 6q^{13} - q^{16} - 2q^{17} - 3q^{18} - 8q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.