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Properties

Label 8736v
Number of curves $4$
Conductor $8736$
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, -9142, -337480]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -9142, -337480]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -9142, -337480]) 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 8736v have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(7\)\(1 - T\)
\(13\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 + 6 T + 11 T^{2}\) 1.11.g
\(17\) \( 1 - 8 T + 17 T^{2}\) 1.17.ai
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 8736v do not have complex multiplication.

Modular form 8736.2.a.v

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

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

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 8736v

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
8736.w3 8736v1 \([0, 1, 0, -9142, -337480]\) \(1320428512222912/9182047329\) \(587651029056\) \([2, 2]\) \(12288\) \(1.0916\) \(\Gamma_0(N)\)-optimal
8736.w1 8736v2 \([0, 1, 0, -146032, -21528052]\) \(672668087746709384/32867289\) \(16828051968\) \([2]\) \(24576\) \(1.4381\)  
8736.w2 8736v3 \([0, 1, 0, -15057, 146367]\) \(92173898928448/50924270943\) \(208585813782528\) \([4]\) \(24576\) \(1.4381\)  
8736.w4 8736v4 \([0, 1, 0, -3472, -745720]\) \(-9043113453704/462519318807\) \(-236809891229184\) \([2]\) \(24576\) \(1.4381\)