Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 0, -135616329, -607843627007]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -135616329, -607843627007]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -135616329, -607843627007]) 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 136710.dd have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 136710.dd do not have complex multiplication.

Modular form 136710.2.a.dd

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);
 
\(q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} + 4 q^{11} - 6 q^{13} + q^{16} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)

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 136710.dd

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
136710.dd1 136710du6 \([1, -1, 0, -135616329, -607843627007]\) \(3216206300355197383681/57660\) \(4945274536860\) \([2]\) \(12582912\) \(2.9041\)  
136710.dd2 136710du4 \([1, -1, 0, -8476029, -9495947147]\) \(785209010066844481/3324675600\) \(285144529795347600\) \([2, 2]\) \(6291456\) \(2.5575\)  
136710.dd3 136710du5 \([1, -1, 0, -8343729, -9806825687]\) \(-749011598724977281/51173462246460\) \(-4388949355018820181660\) \([2]\) \(12582912\) \(2.9041\)  
136710.dd4 136710du3 \([1, -1, 0, -1631709, 625898965]\) \(5601911201812801/1271193750000\) \(109025356976943750000\) \([2]\) \(6291456\) \(2.5575\)  
136710.dd5 136710du2 \([1, -1, 0, -538029, -143395547]\) \(200828550012481/12454560000\) \(1068179299961760000\) \([2, 2]\) \(3145728\) \(2.2109\)  
136710.dd6 136710du1 \([1, -1, 0, 26451, -9387995]\) \(23862997439/457113600\) \(-39204860328345600\) \([2]\) \(1572864\) \(1.8644\) \(\Gamma_0(N)\)-optimal