↑

Properties

Label 436800fa
Number of curves $6$
Conductor $436800$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, 72067, -30335763]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, 72067, -30335763]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, 72067, -30335763]) 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 436800fa have rank \(0\).

Complex multiplication

The elliptic curves in class 436800fa do not have complex multiplication.

Modular form 436800.2.a.fa

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);
 
\(q - q^{3} - q^{7} + q^{9} + 4 q^{11} + q^{13} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

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 436800fa

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
436800.fa6 436800fa1 \([0, -1, 0, 72067, -30335763]\) \(2587063175168/26304786963\) \(-420876591408000000\) \([2]\) \(4718592\) \(2.0629\) \(\Gamma_0(N)\)-optimal*
436800.fa5 436800fa2 \([0, -1, 0, -1128433, -427701263]\) \(620742479063632/49991146569\) \(12797733521664000000\) \([2, 2]\) \(9437184\) \(2.4094\) \(\Gamma_0(N)\)-optimal*
436800.fa4 436800fa3 \([0, -1, 0, -3774433, 2326784737]\) \(5807363790481348/1079211743883\) \(1105112825736192000000\) \([2]\) \(18874368\) \(2.7560\) \(\Gamma_0(N)\)-optimal*
436800.fa2 436800fa4 \([0, -1, 0, -17690433, -28632787263]\) \(597914615076708388/4400862921\) \(4506483631104000000\) \([2, 2]\) \(18874368\) \(2.7560\)  
436800.fa3 436800fa5 \([0, -1, 0, -17326433, -29867839263]\) \(-280880296871140514/25701087819771\) \(-52635827854891008000000\) \([2]\) \(37748736\) \(3.1026\)  
436800.fa1 436800fa6 \([0, -1, 0, -283046433, -1832788231263]\) \(1224522642327678150914/66339\) \(135862272000000\) \([2]\) \(37748736\) \(3.1026\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 436800fa1.