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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 1, -6541, -206341]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -6541, -206341]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -6541, -206341]) 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 510.c have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 + T\)
\(5\)\(1 + T\)
\(17\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 510.c do not have complex multiplication.

Modular form 510.2.a.c

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);
 
\(q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 4 q^{7} + q^{8} + q^{9} - q^{10} - 4 q^{11} - q^{12} - 2 q^{13} - 4 q^{14} + q^{15} + q^{16} + q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

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

Elliptic curves in class 510.c

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
510.c1 510d3 \([1, 1, 1, -6541, -206341]\) \(30949975477232209/478125000\) \(478125000\) \([2]\) \(768\) \(0.80053\)  
510.c2 510d2 \([1, 1, 1, -421, -3157]\) \(8253429989329/936360000\) \(936360000\) \([2, 2]\) \(384\) \(0.45396\)  
510.c3 510d1 \([1, 1, 1, -101, 299]\) \(114013572049/15667200\) \(15667200\) \([4]\) \(192\) \(0.10739\) \(\Gamma_0(N)\)-optimal
510.c4 510d4 \([1, 1, 1, 579, -14757]\) \(21464092074671/109596256200\) \(-109596256200\) \([2]\) \(768\) \(0.80053\)