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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 1, -493186, 133104971]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -493186, 133104971]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -493186, 133104971]) 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 3234.s have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 + T\)
\(7\)\(1\)
\(11\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 4 T + 5 T^{2}\) 1.5.ae
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 3234.s do not have complex multiplication.

Modular form 3234.2.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);
 
\(q + q^{2} - q^{3} + q^{4} + 4 q^{5} - q^{6} + q^{8} + q^{9} + 4 q^{10} + q^{11} - q^{12} - 4 q^{13} - 4 q^{15} + q^{16} + 2 q^{17} + q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labeled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 3234.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
3234.s1 3234s3 \([1, 1, 1, -493186, 133104971]\) \(112763292123580561/1932612\) \(227369869188\) \([2]\) \(36000\) \(1.7208\)  
3234.s2 3234s4 \([1, 1, 1, -492696, 133383291]\) \(-112427521449300721/466873642818\) \(-54927217203894882\) \([2]\) \(72000\) \(2.0674\)  
3234.s3 3234s1 \([1, 1, 1, -2206, -29989]\) \(10091699281/2737152\) \(322023195648\) \([2]\) \(7200\) \(0.91609\) \(\Gamma_0(N)\)-optimal
3234.s4 3234s2 \([1, 1, 1, 5634, -186789]\) \(168105213359/228637728\) \(-26899000061472\) \([2]\) \(14400\) \(1.2627\)