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Properties

Label 489762.dg
Number of curves $6$
Conductor $489762$
CM no
Rank $2$
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, -120362846, 508285978737]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, -120362846, 508285978737]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, -120362846, 508285978737]) 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 489762.dg have rank \(2\).

Complex multiplication

The elliptic curves in class 489762.dg do not have complex multiplication.

Modular form 489762.2.a.dg

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);
 
\(q + q^{2} + q^{4} - 2 q^{5} - q^{7} + q^{8} - 2 q^{10} - 4 q^{11} - q^{14} + q^{16} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 489762.dg

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
489762.dg1 489762dg5 \([1, -1, 1, -120362846, 508285978737]\) \(54804145548726848737/637608031452\) \(2243579282635216770972\) \([2]\) \(75497472\) \(3.2478\) \(\Gamma_0(N)\)-optimal*
489762.dg2 489762dg4 \([1, -1, 1, -26943026, -53820986943]\) \(614716917569296417/19093020912\) \(67183448212742530032\) \([2]\) \(37748736\) \(2.9012\)  
489762.dg3 489762dg3 \([1, -1, 1, -7717586, 7510210881]\) \(14447092394873377/1439452851984\) \(5065065742172356471824\) \([2, 2]\) \(37748736\) \(2.9012\) \(\Gamma_0(N)\)-optimal*
489762.dg4 489762dg2 \([1, -1, 1, -1755266, -765489279]\) \(169967019783457/26337394944\) \(92674544140192944384\) \([2, 2]\) \(18874368\) \(2.5546\) \(\Gamma_0(N)\)-optimal*
489762.dg5 489762dg1 \([1, -1, 1, 191614, -66169983]\) \(221115865823/664731648\) \(-2339020339139248128\) \([2]\) \(9437184\) \(2.2080\) \(\Gamma_0(N)\)-optimal*
489762.dg6 489762dg6 \([1, -1, 1, 9530554, 36307705425]\) \(27207619911317663/177609314617308\) \(-624961667705138627615388\) \([2]\) \(75497472\) \(3.2478\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 489762.dg1.