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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, -635471, -195033699]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -635471, -195033699]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -635471, -195033699]) 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 930.n have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(5\)\(1 + T\)
\(31\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 8 T + 19 T^{2}\) 1.19.ai
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 930.n do not have complex multiplication.

Modular form 930.2.a.n

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

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

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 930.n

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
930.n1 930n4 \([1, 0, 0, -635471, -195033699]\) \(28379906689597370652529/1357352437500\) \(1357352437500\) \([2]\) \(8640\) \(1.8053\)  
930.n2 930n3 \([1, 0, 0, -39651, -3060495]\) \(-6894246873502147249/47925198774000\) \(-47925198774000\) \([2]\) \(4320\) \(1.4587\)  
930.n3 930n2 \([1, 0, 0, -8531, -218655]\) \(68663623745397169/19216056254400\) \(19216056254400\) \([6]\) \(2880\) \(1.2560\)  
930.n4 930n1 \([1, 0, 0, 1389, -22239]\) \(296354077829711/387386634240\) \(-387386634240\) \([6]\) \(1440\) \(0.90944\) \(\Gamma_0(N)\)-optimal