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Properties

Label 33282f
Number of curves $2$
Conductor $33282$
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([1, -1, 0, 74538, -150552716]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, 74538, -150552716]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, 74538, -150552716]) 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 33282f have rank \(0\).

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 \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 4 T + 17 T^{2}\) 1.17.e
\(19\) \( 1 - 8 T + 19 T^{2}\) 1.19.ai
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 33282.2.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);
 
\(q - q^{2} + q^{4} + 2 q^{7} - q^{8} - 5 q^{11} - 2 q^{13} - 2 q^{14} + q^{16} - 7 q^{17} + 7 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

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 33282f

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.h2 33282f1 \([1, -1, 0, 74538, -150552716]\) \(5375/1152\) \(-9815844098731540608\) \([]\) \(809088\) \(2.3234\) \(\Gamma_0(N)\)-optimal
33282.h1 33282f2 \([1, -1, 0, -78637392, 268776627322]\) \(-6311547390625/9565938\) \(-81508468807406073004002\) \([]\) \(5663616\) \(3.2964\)