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Properties

Label 55650bv
Number of curves $4$
Conductor $55650$
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, 87662, -31177969]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, 87662, -31177969]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, 87662, -31177969]) 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 55650bv have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 + T\)
\(5\)\(1\)
\(7\)\(1 + T\)
\(53\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - T + 13 T^{2}\) 1.13.ab
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 55650bv do not have complex multiplication.

Modular form 55650.2.a.bv

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);
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} - q^{7} + q^{8} + q^{9} - q^{12} + 2 q^{13} - q^{14} + q^{16} - 2 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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.

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

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 55650bv

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
55650.bw4 55650bv1 \([1, 1, 1, 87662, -31177969]\) \(4768013769464231/29697948831600\) \(-464030450493750000\) \([2]\) \(737280\) \(2.0717\) \(\Gamma_0(N)\)-optimal
55650.bw3 55650bv2 \([1, 1, 1, -1112838, -410535969]\) \(9754377335041367449/995626517602500\) \(15556664337539062500\) \([2, 2]\) \(1474560\) \(2.4183\)  
55650.bw2 55650bv3 \([1, 1, 1, -4089588, 2732912031]\) \(484108118865316036729/73399966614843750\) \(1146874478356933593750\) \([2]\) \(2949120\) \(2.7649\)  
55650.bw1 55650bv4 \([1, 1, 1, -17344088, -27808885969]\) \(36928196050908253259449/452758954469850\) \(7074358663591406250\) \([2]\) \(2949120\) \(2.7649\)