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Properties

Label 356160es
Number of curves $4$
Conductor $356160$
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, -21201, -1079505]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -21201, -1079505]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -21201, -1079505]) 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 356160es have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1 + T\)
\(7\)\(1 + T\)
\(53\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 8 T + 19 T^{2}\) 1.19.ai
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 356160es do not have complex multiplication.

Modular form 356160.2.a.es

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);
 
\(q + q^{3} - q^{5} - q^{7} + q^{9} + 4 q^{11} + 6 q^{13} - q^{15} + 6 q^{17} + 8 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 356160es

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
356160.es3 356160es1 \([0, 1, 0, -21201, -1079505]\) \(64326999643216/6900761385\) \(113062074531840\) \([2]\) \(1769472\) \(1.4308\) \(\Gamma_0(N)\)-optimal
356160.es2 356160es2 \([0, 1, 0, -79521, 7446879]\) \(848583809246884/122916854025\) \(8055478945382400\) \([2, 2]\) \(3538944\) \(1.7774\)  
356160.es1 356160es3 \([0, 1, 0, -1224321, 521004159]\) \(1548460162842143042/41247151155\) \(5406346596188160\) \([2]\) \(7077888\) \(2.1239\)  
356160.es4 356160es4 \([0, 1, 0, 132159, 40511295]\) \(1947604329004318/6524441476875\) \(-855171593256960000\) \([2]\) \(7077888\) \(2.1239\)