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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 0, -5442, -39034]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -5442, -39034]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -5442, -39034]) 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 450.a have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(5\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(11\) \( 1 + 6 T + 11 T^{2}\) 1.11.g
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 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 450.a do not have complex multiplication.

Modular form 450.2.a.a

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);
 
\(q - q^{2} + q^{4} - 2 q^{7} - q^{8} - 6 q^{11} + 4 q^{13} + 2 q^{14} + q^{16} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

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 450.a

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
450.a1 450f4 \([1, -1, 0, -5442, -39034]\) \(57960603/31250\) \(9610839843750\) \([2]\) \(1152\) \(1.1822\)  
450.a2 450f2 \([1, -1, 0, -3192, 70216]\) \(8527173507/200\) \(84375000\) \([2]\) \(384\) \(0.63293\)  
450.a3 450f1 \([1, -1, 0, -192, 1216]\) \(-1860867/320\) \(-135000000\) \([2]\) \(192\) \(0.28636\) \(\Gamma_0(N)\)-optimal
450.a4 450f3 \([1, -1, 0, 1308, -5284]\) \(804357/500\) \(-153773437500\) \([2]\) \(576\) \(0.83566\)