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Properties

Label 43680.s
Number of curves $4$
Conductor $43680$
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, -39345, 3017025]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -39345, 3017025]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -39345, 3017025]) 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 43680.s have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 43680.s do not have complex multiplication.

Modular form 43680.2.a.s

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);
 
\(q - q^{3} + q^{5} - q^{7} + q^{9} + 4 q^{11} + q^{13} - q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 43680.s

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
43680.s1 43680bl4 \([0, -1, 0, -39345, 3017025]\) \(1644536250830656/4606875\) \(18869760000\) \([4]\) \(81920\) \(1.2038\)  
43680.s2 43680bl3 \([0, -1, 0, -7040, -166428]\) \(75376057236488/19586258055\) \(10028164124160\) \([2]\) \(81920\) \(1.2038\)  
43680.s3 43680bl1 \([0, -1, 0, -2490, 46512]\) \(26688009479104/1358291025\) \(86930625600\) \([2, 2]\) \(40960\) \(0.85719\) \(\Gamma_0(N)\)-optimal
43680.s4 43680bl2 \([0, -1, 0, 1560, 179352]\) \(819500310712/27772859205\) \(-14219703912960\) \([2]\) \(81920\) \(1.2038\)