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Properties

Label 264264.i
Number of curves $6$
Conductor $264264$
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, -342486184, 2439680876140]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -342486184, 2439680876140]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -342486184, 2439680876140]) 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 264264.i have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(7\)\(1 + T\)
\(11\)\(1\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 264264.i do not have complex multiplication.

Modular form 264264.2.a.i

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);
 
\(q - q^{3} - 2 q^{5} - q^{7} + q^{9} - q^{13} + 2 q^{15} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)

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 264264.i

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
264264.i1 264264i5 \([0, -1, 0, -342486184, 2439680876140]\) \(1224522642327678150914/66339\) \(240688302446592\) \([2]\) \(23592960\) \(3.1502\)  
264264.i2 264264i3 \([0, -1, 0, -21405424, 38125223644]\) \(597914615076708388/4400862921\) \(7983510648002233344\) \([2, 2]\) \(11796480\) \(2.8037\)  
264264.i3 264264i6 \([0, -1, 0, -20964984, 39768769548]\) \(-280880296871140514/25701087819771\) \(-93247579830438569023488\) \([2]\) \(23592960\) \(3.1502\)  
264264.i4 264264i4 \([0, -1, 0, -4567064, -3093753540]\) \(5807363790481348/1079211743883\) \(1957774782674034035712\) \([2]\) \(11796480\) \(2.8037\)  
264264.i5 264264i2 \([0, -1, 0, -1365404, 570226164]\) \(620742479063632/49991146569\) \(22671965595372597504\) \([2, 2]\) \(5898240\) \(2.4571\)  
264264.i6 264264i1 \([0, -1, 0, 87201, 40315860]\) \(2587063175168/26304786963\) \(-745608555151347888\) \([2]\) \(2949120\) \(2.1105\) \(\Gamma_0(N)\)-optimal