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Properties

Label 355008.ce
Number of curves $2$
Conductor $355008$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, -9672609601, -412459354507967]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -9672609601, -412459354507967]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -9672609601, -412459354507967]) 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 355008.ce have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(43\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 4 T + 5 T^{2}\) 1.5.ae
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 - T + 11 T^{2}\) 1.11.ab
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 5 T + 17 T^{2}\) 1.17.f
\(19\) \( 1 + 5 T + 19 T^{2}\) 1.19.f
\(23\) \( 1 - 2 T + 23 T^{2}\) 1.23.ac
\(29\) \( 1 - 8 T + 29 T^{2}\) 1.29.ai
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 355008.ce do not have complex multiplication.

Modular form 355008.2.a.ce

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);
 
\(q - q^{3} + 4 q^{5} + 4 q^{7} + q^{9} + q^{11} + 4 q^{13} - 4 q^{15} - 5 q^{17} - 5 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 13 \\ 13 & 1 \end{array}\right)\)

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 355008.ce

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
355008.ce1 355008ce2 \([0, -1, 0, -9672609601, -412459354507967]\) \(-32663831300214001/5083731656658\) \(-15576510958298471391538042109952\) \([]\) \(1406426112\) \(4.7140\)  
355008.ce2 355008ce1 \([0, -1, 0, -157211841, 759107264193]\) \(-140246460241/73728\) \(-225901970736274873516032\) \([]\) \(108186624\) \(3.4315\) \(\Gamma_0(N)\)-optimal