Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 1, -2069779263, -36234533783817]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, -2069779263, -36234533783817]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, -2069779263, -36234533783817]) 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 91358.d have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(17\)\(1 - T\)
\(2687\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + 3 T + 3 T^{2}\) 1.3.d
\(5\) \( 1 + T + 5 T^{2}\) 1.5.b
\(7\) \( 1 - T + 7 T^{2}\) 1.7.ab
\(11\) \( 1 + 2 T + 11 T^{2}\) 1.11.c
\(13\) \( 1 + 7 T + 13 T^{2}\) 1.13.h
\(19\) \( 1 + 8 T + 19 T^{2}\) 1.19.i
\(23\) \( 1 - 3 T + 23 T^{2}\) 1.23.ad
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 91358.d do not have complex multiplication.

Modular form 91358.2.a.d

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

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

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 91358.d

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
91358.d1 91358e2 \([1, -1, 1, -2069779263, -36234533783817]\) \(980608845198487572940262772638529/275069582998987905611470576\) \(275069582998987905611470576\) \([]\) \(130658304\) \(4.0546\)  
91358.d2 91358e1 \([1, -1, 1, -62268303, 189138631383]\) \(26700832091472147512881229889/295971568928797229056\) \(295971568928797229056\) \([7]\) \(18665472\) \(3.0817\) \(\Gamma_0(N)\)-optimal