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Properties

Label 356160.bh
Number of curves $4$
Conductor $356160$
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([0, -1, 0, -4092920641, 100784633055841]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -4092920641, 100784633055841]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -4092920641, 100784633055841]) 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.bh have rank \(0\).

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.c
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 8 T + 19 T^{2}\) 1.19.i
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 356160.2.a.bh

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);
 
\(q - q^{3} - q^{5} + q^{7} + q^{9} - 2 q^{13} + q^{15} - 6 q^{17} - 8 q^{19} + O(q^{20})\) 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 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.bh

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.bh1 356160bh3 \([0, -1, 0, -4092920641, 100784633055841]\) \(115702769301750288368222218564/2978542501586017340625\) \(195201761383941232435200000\) \([2]\) \(355532800\) \(4.1516\)  
356160.bh2 356160bh2 \([0, -1, 0, -265670641, 1446828405841]\) \(126571074009076580447874256/18059144285999619140625\) \(295881019981817760000000000\) \([2, 2]\) \(177766400\) \(3.8050\)  
356160.bh3 356160bh1 \([0, -1, 0, -70358141, -204616906659]\) \(37615499197072174665988096/4101083850860595703125\) \(4199509863281250000000000\) \([2]\) \(88883200\) \(3.4585\) \(\Gamma_0(N)\)-optimal
356160.bh4 356160bh4 \([0, -1, 0, 436579359, 7798398755841]\) \(140421425391704081123281436/482157640388659572028125\) \(-31598683120511193712435200000\) \([2]\) \(355532800\) \(4.1516\)