Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 1, 1, -11, 11]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 1, -11, 11]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 1, -11, 11]) 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 435.c have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1 - T\)
\(5\)\(1 + T\)
\(29\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + 2 T^{2}\) 1.2.a
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(11\) \( 1 - 3 T + 11 T^{2}\) 1.11.ad
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 - 3 T + 23 T^{2}\) 1.23.ad
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 435.c do not have complex multiplication.

Modular form 435.2.a.c

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);
 
\(q + q^{3} - 2 q^{4} - q^{5} + 2 q^{7} + q^{9} + 3 q^{11} - 2 q^{12} + 2 q^{13} - q^{15} + 4 q^{16} + 2 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labeled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 435.c

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
435.c1 435a1 \([0, 1, 1, -11, 11]\) \(-160989184/3915\) \(-3915\) \([3]\) \(20\) \(-0.52517\) \(\Gamma_0(N)\)-optimal
435.c2 435a2 \([0, 1, 1, 49, 80]\) \(12747309056/9145875\) \(-9145875\) \([]\) \(60\) \(0.024133\)