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Properties

Label 33282bg
Number of curves $2$
Conductor $33282$
CM no
Rank $2$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 1, -11957, 506445]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, -11957, 506445]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, -11957, 506445]) 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 33282bg have rank \(2\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(43\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 3 T + 5 T^{2}\) 1.5.ad
\(7\) \( 1 - T + 7 T^{2}\) 1.7.ab
\(11\) \( 1 - T + 11 T^{2}\) 1.11.ab
\(13\) \( 1 - T + 13 T^{2}\) 1.13.ab
\(17\) \( 1 + 4 T + 17 T^{2}\) 1.17.e
\(19\) \( 1 + T + 19 T^{2}\) 1.19.b
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 + 9 T + 29 T^{2}\) 1.29.j
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 33282bg do not have complex multiplication.

Modular form 33282.2.a.bg

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);
 
\(q + q^{2} + q^{4} - 4 q^{5} - 4 q^{7} + q^{8} - 4 q^{10} + q^{11} - 4 q^{13} - 4 q^{14} + q^{16} + 5 q^{17} - 5 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 13 \\ 13 & 1 \end{array}\right)\)

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 33282bg

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
33282.q2 33282bg1 \([1, -1, 1, -11957, 506445]\) \(-140246460241/73728\) \(-99379519488\) \([]\) \(104832\) \(1.0605\) \(\Gamma_0(N)\)-optimal
33282.q1 33282bg2 \([1, -1, 1, -735647, -273388935]\) \(-32663831300214001/5083731656658\) \(-6852468658374108018\) \([]\) \(1362816\) \(2.3430\)