Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, -1, 0]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -1, 0]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -1, 0]) 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 65.a have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(5\)\(1 + T\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + T + 2 T^{2}\) 1.2.b
\(3\) \( 1 + 2 T + 3 T^{2}\) 1.3.c
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(11\) \( 1 - 2 T + 11 T^{2}\) 1.11.ac
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 6 T + 19 T^{2}\) 1.19.g
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 65.a do not have complex multiplication.

Modular form 65.2.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);
 
\(q - q^{2} - 2 q^{3} - q^{4} - q^{5} + 2 q^{6} - 4 q^{7} + 3 q^{8} + q^{9} + q^{10} + 2 q^{11} + 2 q^{12} - q^{13} + 4 q^{14} + 2 q^{15} - q^{16} + 2 q^{17} - q^{18} - 6 q^{19} + O(q^{20})\) 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.

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

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 65.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
65.a1 65a1 \([1, 0, 0, -1, 0]\) \(117649/65\) \(65\) \([2]\) \(2\) \(-0.96162\) \(\Gamma_0(N)\)-optimal
65.a2 65a2 \([1, 0, 0, 4, 1]\) \(6967871/4225\) \(-4225\) \([2]\) \(4\) \(-0.61504\)