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Properties

Label 13104w
Number of curves $6$
Conductor $13104$
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, 0, 0, 6486, -818417]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, 6486, -818417]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, 6486, -818417]) 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 13104w have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 13104w do not have complex multiplication.

Modular form 13104.2.a.w

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);
 
\(q + 2 q^{5} - q^{7} - 4 q^{11} + q^{13} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

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 13104w

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
13104.bv6 13104w1 \([0, 0, 0, 6486, -818417]\) \(2587063175168/26304786963\) \(-306819035136432\) \([2]\) \(36864\) \(1.4609\) \(\Gamma_0(N)\)-optimal
13104.bv5 13104w2 \([0, 0, 0, -101559, -11558090]\) \(620742479063632/49991146569\) \(9329547737293056\) \([2, 2]\) \(73728\) \(1.8074\)  
13104.bv2 13104w3 \([0, 0, 0, -1592139, -773244470]\) \(597914615076708388/4400862921\) \(3285226567074816\) \([2, 2]\) \(147456\) \(2.1540\)  
13104.bv4 13104w4 \([0, 0, 0, -339699, 62789218]\) \(5807363790481348/1079211743883\) \(805627249961683968\) \([4]\) \(147456\) \(2.1540\)  
13104.bv1 13104w5 \([0, 0, 0, -25474179, -49487829662]\) \(1224522642327678150914/66339\) \(99043596288\) \([2]\) \(294912\) \(2.5006\)  
13104.bv3 13104w6 \([0, 0, 0, -1559379, -806587598]\) \(-280880296871140514/25701087819771\) \(-38371518506215544832\) \([2]\) \(294912\) \(2.5006\)