↑

Properties

Label 446490z
Number of curves $2$
Conductor $446490$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 0, -330252885, -2314901348325]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -330252885, -2314901348325]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -330252885, -2314901348325]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 446490z have rank \(1\).

Complex multiplication

The elliptic curves in class 446490z do not have complex multiplication.

Modular form 446490.2.a.z

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} + q^{10} + 4 q^{13} - q^{14} + q^{16} - 5 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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 comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 446490z

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
446490.z2 446490z1 \([1, -1, 0, -330252885, -2314901348325]\) \(-3084465621865350349801/7660617292968750\) \(-9893441856636629542968750\) \([]\) \(138516480\) \(3.6739\) \(\Gamma_0(N)\)-optimal
446490.z1 446490z2 \([1, -1, 0, -26766408510, -1685510810183700]\) \(-1642140750896013526054039801/25018323000\) \(-32310362792595987000\) \([]\) \(415549440\) \(4.2233\)