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Properties

Label 499800cz
Number of curves $2$
Conductor $499800$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cz1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 499800cz have rank \(0\).

Complex multiplication

The elliptic curves in class 499800cz do not have complex multiplication.

Modular form 499800.2.a.cz

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 499800cz

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
499800.cz2 499800cz1 \([0, -1, 0, -776356408, 12742543862812]\) \(-27491530342319084164/21352892495484375\) \(-40194343187219859750000000000\) \([2]\) \(371589120\) \(4.1874\) \(\Gamma_0(N)\)-optimal*
499800.cz1 499800cz2 \([0, -1, 0, -14171731408, 649210391612812]\) \(83609231549925663172082/22317062975463375\) \(84018564544009299372000000000\) \([2]\) \(743178240\) \(4.5339\) \(\Gamma_0(N)\)-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 499800cz1.