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Properties

Label 445200.gz
Number of curves $4$
Conductor $445200$
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, -277505408, 1779213691188]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -277505408, 1779213691188]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -277505408, 1779213691188]) 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 445200.gz have rank \(1\).

Complex multiplication

The elliptic curves in class 445200.gz do not have complex multiplication.

Modular form 445200.2.a.gz

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);
 
\(q + q^{3} + q^{7} + q^{9} + 2 q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

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 445200.gz

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
445200.gz1 445200gz3 \([0, 1, 0, -277505408, 1779213691188]\) \(36928196050908253259449/452758954469850\) \(28976573086070400000000\) \([2]\) \(70778880\) \(3.4580\) \(\Gamma_0(N)\)-optimal*
445200.gz2 445200gz4 \([0, 1, 0, -65433408, -175037236812]\) \(484108118865316036729/73399966614843750\) \(4697597863350000000000000\) \([2]\) \(70778880\) \(3.4580\)  
445200.gz3 445200gz2 \([0, 1, 0, -17805408, 26238691188]\) \(9754377335041367449/995626517602500\) \(63720097126560000000000\) \([2, 2]\) \(35389440\) \(3.1115\) \(\Gamma_0(N)\)-optimal*
445200.gz4 445200gz1 \([0, 1, 0, 1402592, 1998195188]\) \(4768013769464231/29697948831600\) \(-1900668725222400000000\) \([2]\) \(17694720\) \(2.7649\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 445200.gz1.