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Properties

Label 44520.q
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, -157736, 24057264]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -157736, 24057264]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -157736, 24057264]) 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 44520.q 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 \(\mathbb{F}_p\)
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 44520.2.a.q

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);
 
\(q + q^{3} - q^{5} - q^{7} + q^{9} - 2 q^{13} - q^{15} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 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 44520.q

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.q1 44520g4 \([0, 1, 0, -157736, 24057264]\) \(211928314150936658/28997517675\) \(59386916198400\) \([2]\) \(196608\) \(1.6609\)  
44520.q2 44520g3 \([0, 1, 0, -63736, -5973136]\) \(13981539250684658/618707267325\) \(1267112483481600\) \([2]\) \(196608\) \(1.6609\)  
44520.q3 44520g2 \([0, 1, 0, -10736, 302064]\) \(133657002021316/37937300625\) \(38847795840000\) \([2, 2]\) \(98304\) \(1.3143\)  
44520.q4 44520g1 \([0, 1, 0, 1764, 32064]\) \(2369917714736/3043359375\) \(-779100000000\) \([2]\) \(49152\) \(0.96773\) \(\Gamma_0(N)\)-optimal