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Properties

Label 50575bc
Number of curves $2$
Conductor $50575$
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, 1, 482, -5526]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 1, 482, -5526]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 1, 482, -5526]) 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 50575bc have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(5\)\(1\)
\(7\)\(1 - T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(3\) \( 1 - 2 T + 3 T^{2}\) 1.3.ac
\(11\) \( 1 + 3 T + 11 T^{2}\) 1.11.d
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 3 T + 23 T^{2}\) 1.23.d
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 50575bc do not have complex multiplication.

Modular form 50575.2.a.bc

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);
 
\(q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{6} - q^{7} - 2 q^{9} + 3 q^{11} + 2 q^{12} - q^{13} + 2 q^{14} - 4 q^{16} + 4 q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

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 50575bc

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
50575.a2 50575bc1 \([0, 1, 1, 482, -5526]\) \(4096/7\) \(-21120372875\) \([]\) \(35200\) \(0.66608\) \(\Gamma_0(N)\)-optimal
50575.a1 50575bc2 \([0, 1, 1, -42868, 3419124]\) \(-2887553024/16807\) \(-50710015272875\) \([]\) \(176000\) \(1.4708\)