Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 0, -11, -27]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -11, -27]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -11, -27]) 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 3234.g have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(7\)\(1\)
\(11\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 3 T + 5 T^{2}\) 1.5.ad
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 3 T + 17 T^{2}\) 1.17.d
\(19\) \( 1 + 2 T + 19 T^{2}\) 1.19.c
\(23\) \( 1 - 3 T + 23 T^{2}\) 1.23.ad
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 3234.g do not have complex multiplication.

Modular form 3234.2.a.g

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);
 
\(q - q^{2} - q^{3} + q^{4} + 3 q^{5} + q^{6} - q^{8} + q^{9} - 3 q^{10} - q^{11} - q^{12} - 2 q^{13} - 3 q^{15} + q^{16} - 3 q^{17} - q^{18} - 2 q^{19} + O(q^{20})\) 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 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 3234.g

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
3234.g1 3234d1 \([1, 1, 0, -11, -27]\) \(-3451273/2376\) \(-116424\) \([]\) \(432\) \(-0.32759\) \(\Gamma_0(N)\)-optimal
3234.g2 3234d2 \([1, 1, 0, 94, 372]\) \(1843623047/2044416\) \(-100176384\) \([]\) \(1296\) \(0.22171\)