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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 1, -6935, -36241]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, -6935, -36241]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, -6935, -36241]) 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 414.c have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(23\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 414.c do not have complex multiplication.

Modular form 414.2.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);
 
\(q + q^{2} + q^{4} + 2 q^{7} + q^{8} + 2 q^{13} + 2 q^{14} + q^{16} + 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

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 414.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
414.c1 414a4 \([1, -1, 1, -6935, -36241]\) \(50591419971625/28422890688\) \(20720287311552\) \([6]\) \(768\) \(1.2453\)  
414.c2 414a2 \([1, -1, 1, -5180, -142189]\) \(21081759765625/57132\) \(41649228\) \([2]\) \(256\) \(0.69601\)  
414.c3 414a1 \([1, -1, 1, -320, -2221]\) \(-4956477625/268272\) \(-195570288\) \([2]\) \(128\) \(0.34944\) \(\Gamma_0(N)\)-optimal
414.c4 414a3 \([1, -1, 1, 1705, -5137]\) \(752329532375/448524288\) \(-326974205952\) \([6]\) \(384\) \(0.89874\)