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Properties

Label 491970.ci
Number of curves $6$
Conductor $491970$
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, 0, 0, -162678091, 798609261845]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -162678091, 798609261845]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -162678091, 798609261845]) 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 491970.ci have rank \(0\).

Complex multiplication

The elliptic curves in class 491970.ci do not have complex multiplication.

Modular form 491970.2.a.ci

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

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

Isogeny graph

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

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

Elliptic curves in class 491970.ci

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
491970.ci1 491970ci5 \([1, 0, 0, -162678091, 798609261845]\) \(3216206300355197383681/57660\) \(8535749359740\) \([2]\) \(46137344\) \(2.9496\) \(\Gamma_0(N)\)-optimal*
491970.ci2 491970ci3 \([1, 0, 0, -10167391, 12477607625]\) \(785209010066844481/3324675600\) \(492171308082608400\) \([2, 2]\) \(23068672\) \(2.6030\) \(\Gamma_0(N)\)-optimal*
491970.ci3 491970ci6 \([1, 0, 0, -10008691, 12886006205]\) \(-749011598724977281/51173462246460\) \(-7575508976862643202940\) \([2]\) \(46137344\) \(2.9496\)  
491970.ci4 491970ci4 \([1, 0, 0, -1957311, -821926359]\) \(5601911201812801/1271193750000\) \(188182296872493750000\) \([2]\) \(23068672\) \(2.6030\)  
491970.ci5 491970ci2 \([1, 0, 0, -645391, 188514425]\) \(200828550012481/12454560000\) \(1843721861703840000\) \([2, 2]\) \(11534336\) \(2.2564\) \(\Gamma_0(N)\)-optimal*
491970.ci6 491970ci1 \([1, 0, 0, 31729, 12327801]\) \(23862997439/457113600\) \(-67669218149990400\) \([2]\) \(5767168\) \(1.9099\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 491970.ci1.