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Properties

Label 356160.bz
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, -523909505, -4367019529215]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -523909505, -4367019529215]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -523909505, -4367019529215]) 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 356160.bz 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.e
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 356160.2.a.bz

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);
 
\(q - q^{3} + q^{5} - q^{7} + q^{9} - 4 q^{11} + 6 q^{13} - q^{15} - 2 q^{17} + O(q^{20})\) 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.

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

Isogeny graph

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

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

Elliptic curves in class 356160.bz

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.bz1 356160bz3 \([0, -1, 0, -523909505, -4367019529215]\) \(242668058425926391337530756/14711617123833656848815\) \(964140539827562535243939840\) \([2]\) \(176357376\) \(3.9301\)  
356160.bz2 356160bz2 \([0, -1, 0, -516036305, -4511806102575]\) \(927565408657759969339679824/3744744159144881025\) \(61353888303429730713600\) \([2, 2]\) \(88178688\) \(3.5835\)  
356160.bz3 356160bz1 \([0, -1, 0, -516035805, -4511815283475]\) \(14841003399024074060155869184/241891768125\) \(247697170560000\) \([2]\) \(44089344\) \(3.2369\) \(\Gamma_0(N)\)-optimal
356160.bz4 356160bz4 \([0, -1, 0, -508171105, -4656005106335]\) \(-221448979693296464284621156/14756554069224468581115\) \(-967085527480694772931952640\) \([2]\) \(176357376\) \(3.9301\)