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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, -12309, 529757]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -12309, 529757]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -12309, 529757]) 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 304.f have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(19\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - 2 T + 3 T^{2}\) 1.3.ac
\(5\) \( 1 - 3 T + 5 T^{2}\) 1.5.ad
\(7\) \( 1 - T + 7 T^{2}\) 1.7.ab
\(11\) \( 1 + 3 T + 11 T^{2}\) 1.11.d
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 + 3 T + 17 T^{2}\) 1.17.d
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 304.f do not have complex multiplication.

Modular form 304.2.a.f

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

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 304.f

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
304.f1 304e3 \([0, -1, 0, -12309, 529757]\) \(-50357871050752/19\) \(-77824\) \([]\) \(216\) \(0.72659\)  
304.f2 304e2 \([0, -1, 0, -149, 797]\) \(-89915392/6859\) \(-28094464\) \([]\) \(72\) \(0.17728\)  
304.f3 304e1 \([0, -1, 0, 11, -3]\) \(32768/19\) \(-77824\) \([]\) \(24\) \(-0.37203\) \(\Gamma_0(N)\)-optimal