↑

Properties

Label 486330cd
Number of curves $2$
Conductor $486330$
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, 0, 0, 2098150, 1610422500]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, 2098150, 1610422500]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, 2098150, 1610422500]) 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 486330cd have rank \(1\).

Complex multiplication

The elliptic curves in class 486330cd do not have complex multiplication.

Modular form 486330.2.a.cd

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);
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + q^{10} - 2 q^{11} + q^{12} + q^{13} + q^{14} + q^{15} + q^{16} + 4 q^{17} + q^{18} - 8 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

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 486330cd

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
486330.cd2 486330cd1 \([1, 0, 0, 2098150, 1610422500]\) \(1021488070932591813813599/1711270653287130000000\) \(-1711270653287130000000\) \([7]\) \(29042496\) \(2.7593\) \(\Gamma_0(N)\)-optimal
486330.cd1 486330cd2 \([1, 0, 0, -716997500, -7390008178230]\) \(-40763985302355967860626447640001/1828646593831170189984570\) \(-1828646593831170189984570\) \([]\) \(203297472\) \(3.7323\)