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Properties

Label 470592cj
Number of curves $4$
Conductor $470592$
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([0, 0, 0, -3456300, -2468306896]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -3456300, -2468306896]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -3456300, -2468306896]) 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 470592cj have rank \(0\).

Complex multiplication

The elliptic curves in class 470592cj do not have complex multiplication.

Modular form 470592.2.a.cj

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);
 
\(q - 4 q^{7} - 2 q^{13} + 6 q^{17} - q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

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 470592cj

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
470592.cj2 470592cj1 \([0, 0, 0, -3456300, -2468306896]\) \(23894093340015625/55042322688\) \(10518751671629119488\) \([2]\) \(10616832\) \(2.5310\) \(\Gamma_0(N)\)-optimal*
470592.cj3 470592cj2 \([0, 0, 0, -2212140, -4269352912]\) \(-6264610702863625/37578744274608\) \(-7181409865220550033408\) \([2]\) \(21233664\) \(2.8775\)  
470592.cj1 470592cj3 \([0, 0, 0, -15768300, 21907286192]\) \(2268876641163765625/228097945239552\) \(43590196154763420106752\) \([2]\) \(31850496\) \(3.0803\) \(\Gamma_0(N)\)-optimal*
470592.cj4 470592cj4 \([0, 0, 0, 19621140, 106233243824]\) \(4371484788393482375/28041364201746432\) \(-5358788150053607552581632\) \([2]\) \(63700992\) \(3.4269\)  
*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 470592cj1.