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Properties

Label 33282a
Number of curves $2$
Conductor $33282$
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([1, -1, 0, -163983, 23011065]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -163983, 23011065]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -163983, 23011065]) 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 33282a have rank \(1\).

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.d
\(7\) \( 1 - 3 T + 7 T^{2}\) 1.7.ad
\(11\) \( 1 - 5 T + 11 T^{2}\) 1.11.af
\(13\) \( 1 + 3 T + 13 T^{2}\) 1.13.d
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 + 7 T + 19 T^{2}\) 1.19.h
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 - T + 29 T^{2}\) 1.29.ab
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 33282.2.a.a

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

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

Isogeny graph

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

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

Elliptic curves in class 33282a

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.f2 33282a1 \([1, -1, 0, -163983, 23011065]\) \(35937/4\) \(54280002089179044\) \([2]\) \(302720\) \(1.9442\) \(\Gamma_0(N)\)-optimal
33282.f1 33282a2 \([1, -1, 0, -2549193, 1567196019]\) \(135005697/2\) \(27140001044589522\) \([2]\) \(605440\) \(2.2907\)