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Properties

Label 436800jt
Number of curves $6$
Conductor $436800$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, -600033, 178979937]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -600033, 178979937]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -600033, 178979937]) 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 436800jt have rank \(1\).

Complex multiplication

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

Modular form 436800.2.a.jt

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);
 
\(q - q^{3} + q^{7} + q^{9} + 4 q^{11} + q^{13} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 436800jt

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.jt5 436800jt1 \([0, -1, 0, -600033, 178979937]\) \(5832972054001/4542720\) \(18606981120000000\) \([2]\) \(4718592\) \(2.0532\) \(\Gamma_0(N)\)-optimal
436800.jt4 436800jt2 \([0, -1, 0, -728033, 97187937]\) \(10418796526321/5038160400\) \(20636304998400000000\) \([2, 2]\) \(9437184\) \(2.3997\)  
436800.jt6 436800jt3 \([0, -1, 0, 2631967, 738947937]\) \(492271755328079/342606902820\) \(-1403317873950720000000\) \([2]\) \(18874368\) \(2.7463\)  
436800.jt2 436800jt4 \([0, -1, 0, -6136033, -5781308063]\) \(6237734630203441/82168222500\) \(336561039360000000000\) \([2, 2]\) \(18874368\) \(2.7463\)  
436800.jt3 436800jt5 \([0, -1, 0, -936033, -15271308063]\) \(-22143063655441/24584858584650\) \(-100699580762726400000000\) \([2]\) \(37748736\) \(3.0929\)  
436800.jt1 436800jt6 \([0, -1, 0, -97864033, -372601580063]\) \(25306558948218234961/4478906250\) \(18345600000000000000\) \([2]\) \(37748736\) \(3.0929\)