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Properties

Label 405720p
Number of curves $2$
Conductor $405720$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 0, -2824563, 1644588862]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -2824563, 1644588862]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -2824563, 1644588862]) 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 405720p have rank \(1\).

Complex multiplication

The elliptic curves in class 405720p do not have complex multiplication.

Modular form 405720.2.a.p

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);
 
\(q - q^{5} - 4 q^{11} + 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

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 405720p

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
405720.p2 405720p1 \([0, 0, 0, -2824563, 1644588862]\) \(9733205763526108/1069365234375\) \(273808966410000000000\) \([2]\) \(13271040\) \(2.6551\) \(\Gamma_0(N)\)-optimal
405720.p1 405720p2 \([0, 0, 0, -10699563, -11706686138]\) \(264527137402013054/37471584403125\) \(19189058079228307200000\) \([2]\) \(26542080\) \(3.0016\)