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Properties

Label 133560.f
Number of curves $4$
Conductor $133560$
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, 0, 0, -9209071443, 340143532027742]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -9209071443, 340143532027742]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -9209071443, 340143532027742]) 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 133560.f have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1 + T\)
\(7\)\(1 + T\)
\(53\)\(1 - T\)
 
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
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 8 T + 19 T^{2}\) 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 133560.f do not have complex multiplication.

Modular form 133560.2.a.f

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);
 
\(q - q^{5} - q^{7} + 2 q^{13} + 6 q^{17} - 8 q^{19} + O(q^{20})\) 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 133560.f

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
133560.f1 133560l3 \([0, 0, 0, -9209071443, 340143532027742]\) \(115702769301750288368222218564/2978542501586017340625\) \(2223470063263955600707200000\) \([2]\) \(177766400\) \(4.3543\)  
133560.f2 133560l2 \([0, 0, 0, -597758943, 4882746990242]\) \(126571074009076580447874256/18059144285999619140625\) \(3370269743230392922500000000\) \([2, 2]\) \(88883200\) \(4.0078\)  
133560.f3 133560l1 \([0, 0, 0, -158305818, -690661212883]\) \(37615499197072174665988096/4101083850860595703125\) \(47835042036437988281250000\) \([2]\) \(44441600\) \(3.6612\) \(\Gamma_0(N)\)-optimal
133560.f4 133560l4 \([0, 0, 0, 982303557, 26320086952742]\) \(140421425391704081123281436/482157640388659572028125\) \(-359928749919572815880707200000\) \([2]\) \(177766400\) \(4.3543\)