Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 1, -33, 68]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, -33, 68]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, -33, 68]) 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 130a have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 130a do not have complex multiplication.

Modular form 130.2.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);
 
\(q - q^{2} - 2 q^{3} + q^{4} + q^{5} + 2 q^{6} - 4 q^{7} - q^{8} + q^{9} - q^{10} - 6 q^{11} - 2 q^{12} + q^{13} + 4 q^{14} - 2 q^{15} + q^{16} - 6 q^{17} - q^{18} + 2 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 130a

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
130.a2 130a1 \([1, 0, 1, -33, 68]\) \(3803721481/26000\) \(26000\) \([6]\) \(24\) \(-0.31886\) \(\Gamma_0(N)\)-optimal
130.a3 130a2 \([1, 0, 1, -13, 156]\) \(-217081801/10562500\) \(-10562500\) \([6]\) \(48\) \(0.027717\)  
130.a1 130a3 \([1, 0, 1, -208, -1122]\) \(988345570681/44994560\) \(44994560\) \([2]\) \(72\) \(0.23045\)  
130.a4 130a4 \([1, 0, 1, 112, -4194]\) \(157376536199/7722894400\) \(-7722894400\) \([2]\) \(144\) \(0.57702\)