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Properties

Label 485775k
Number of curves $2$
Conductor $485775$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 1, -680, -31678]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, -680, -31678]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, -680, -31678]) 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 485775k have rank \(0\).

Complex multiplication

The elliptic curves in class 485775k do not have complex multiplication.

Modular form 485775.2.a.k

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);
 
\(q - q^{2} - q^{4} + 2 q^{7} + 3 q^{8} - 4 q^{11} - 2 q^{13} - 2 q^{14} - q^{16} - q^{17} + 4 q^{19} + O(q^{20})\) 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 485775k

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
485775.k2 485775k1 \([1, -1, 1, -680, -31678]\) \(-3048625/36703\) \(-418070109375\) \([2]\) \(470016\) \(0.91174\) \(\Gamma_0(N)\)-optimal*
485775.k1 485775k2 \([1, -1, 1, -19805, -1064428]\) \(75418890625/274193\) \(3123229640625\) \([2]\) \(940032\) \(1.2583\) \(\Gamma_0(N)\)-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 485775k1.