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Properties

Label 5577d
Number of curves $4$
Conductor $5577$
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, 1, -1102, -14422]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -1102, -14422]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -1102, -14422]) 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 5577d have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1 + T\)
\(11\)\(1 - T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + 2 T^{2}\) 1.2.a
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(7\) \( 1 + 3 T + 7 T^{2}\) 1.7.d
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 5577d do not have complex multiplication.

Modular form 5577.2.a.d

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);
 
\(q - q^{2} - q^{3} - q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 3 q^{8} + q^{9} - 2 q^{10} - q^{11} + q^{12} + 4 q^{14} - 2 q^{15} - q^{16} - 2 q^{17} - q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 5577d

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
5577.a3 5577d1 \([1, 1, 1, -1102, -14422]\) \(30664297/297\) \(1433562273\) \([2]\) \(3456\) \(0.57670\) \(\Gamma_0(N)\)-optimal
5577.a2 5577d2 \([1, 1, 1, -1947, 9576]\) \(169112377/88209\) \(425767995081\) \([2, 2]\) \(6912\) \(0.92327\)  
5577.a1 5577d3 \([1, 1, 1, -24762, 1487988]\) \(347873904937/395307\) \(1908071385363\) \([2]\) \(13824\) \(1.2698\)  
5577.a4 5577d4 \([1, 1, 1, 7348, 83936]\) \(9090072503/5845851\) \(-28216806219459\) \([2]\) \(13824\) \(1.2698\)