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Properties

Label 13475.e
Number of curves $2$
Conductor $13475$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 1, -1599263, 777776656]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -1599263, 777776656]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -1599263, 777776656]) 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 13475.e have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(5\)\(1\)
\(7\)\(1\)
\(11\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + T + 2 T^{2}\) 1.2.b
\(3\) \( 1 - 2 T + 3 T^{2}\) 1.3.ac
\(13\) \( 1 + 6 T + 13 T^{2}\) 1.13.g
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 13475.e do not have complex multiplication.

Modular form 13475.2.a.e

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

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 13475.e

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
13475.e1 13475v2 \([1, 1, 1, -1599263, 777776656]\) \(1968634623437/5929\) \(1362384611328125\) \([2]\) \(199680\) \(2.1318\)  
13475.e2 13475v1 \([1, 1, 1, -98638, 12457906]\) \(-461889917/26411\) \(-6068804177734375\) \([2]\) \(99840\) \(1.7853\) \(\Gamma_0(N)\)-optimal