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Properties

Label 453882z
Number of curves $2$
Conductor $453882$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("z1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 453882z have rank \(0\).

Complex multiplication

The elliptic curves in class 453882z do not have complex multiplication.

Modular form 453882.2.a.z

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - 3 q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + 3 q^{10} - q^{11} + q^{12} + q^{13} - q^{14} - 3 q^{15} + q^{16} - q^{18} - 2 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 453882z

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
453882.z1 453882z1 \([1, 0, 1, -55008870, -157039962440]\) \(-124352595912593543977/103332962304\) \(-15296986937676128256\) \([]\) \(38918880\) \(2.9845\) \(\Gamma_0(N)\)-optimal*
453882.z2 453882z2 \([1, 0, 1, -42701685, -229154829440]\) \(-58169016237585194137/119573538788081664\) \(-17701175115369651734839296\) \([]\) \(116756640\) \(3.5338\) \(\Gamma_0(N)\)-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 453882z1.