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Properties

Label 44520x
Number of curves $4$
Conductor $44520$
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, 1, 0, -5300, 132288]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -5300, 132288]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -5300, 132288]) 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 44520x have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(7\)\(1 - T\)
\(53\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 + 6 T + 13 T^{2}\) 1.13.g
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 8 T + 19 T^{2}\) 1.19.ai
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 44520x do not have complex multiplication.

Modular form 44520.2.a.x

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);
 
\(q + q^{3} + q^{5} + q^{7} + q^{9} + 4 q^{11} - 6 q^{13} + q^{15} + 6 q^{17} + 8 q^{19} + O(q^{20})\) 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.

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

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 44520x

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
44520.x3 44520x1 \([0, 1, 0, -5300, 132288]\) \(64326999643216/6900761385\) \(1766594914560\) \([4]\) \(110592\) \(1.0842\) \(\Gamma_0(N)\)-optimal
44520.x2 44520x2 \([0, 1, 0, -19880, -940800]\) \(848583809246884/122916854025\) \(125866858521600\) \([2, 2]\) \(221184\) \(1.4308\)  
44520.x4 44520x3 \([0, 1, 0, 33040, -5047392]\) \(1947604329004318/6524441476875\) \(-13362056144640000\) \([2]\) \(442368\) \(1.7774\)  
44520.x1 44520x4 \([0, 1, 0, -306080, -65278560]\) \(1548460162842143042/41247151155\) \(84474165565440\) \([2]\) \(442368\) \(1.7774\)