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Properties

Label 356160gm
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, 224415, 127929375]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, 224415, 127929375]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, 224415, 127929375]) 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 356160gm 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 \(\mathbb{F}_p\)
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 356160.2.a.gm

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);
 
\(q + q^{3} + q^{5} - q^{7} + q^{9} + 2 q^{13} + q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 356160gm

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.gm4 356160gm1 \([0, 1, 0, 224415, 127929375]\) \(4768013769464231/29697948831600\) \(-7785139098510950400\) \([2]\) \(5898240\) \(2.3067\) \(\Gamma_0(N)\)-optimal
356160.gm3 356160gm2 \([0, 1, 0, -2848865, 1678706463]\) \(9754377335041367449/995626517602500\) \(260997517830389760000\) \([2, 2]\) \(11796480\) \(2.6533\)  
356160.gm1 356160gm3 \([0, 1, 0, -44400865, 113860796063]\) \(36928196050908253259449/452758954469850\) \(118688043360544358400\) \([4]\) \(23592960\) \(2.9999\)  
356160.gm2 356160gm4 \([0, 1, 0, -10469345, -11204477025]\) \(484108118865316036729/73399966614843750\) \(19241360848281600000000\) \([2]\) \(23592960\) \(2.9999\)