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Properties

Label 13650bz
Number of curves $4$
Conductor $13650$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bz1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 13650bz have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 + T\)
\(5\)\(1\)
\(7\)\(1 - T\)
\(13\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 13650bz do not have complex multiplication.

Modular form 13650.2.a.bz

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} + q^{7} + q^{8} + q^{9} - 4 q^{11} - q^{12} + q^{13} + q^{14} + q^{16} + 2 q^{17} + q^{18} + 8 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content sage:E.isogeny_class().matrix()
 

The \(i,j\) 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 sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.

Elliptic curves in class 13650bz

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13650.ca3 13650bz1 \([1, 1, 1, -3472253588, 78751157917781]\) \(296304326013275547793071733369/268420373544960000000\) \(4194068336640000000000000\) \([4]\) \(10321920\) \(4.0229\) \(\Gamma_0(N)\)-optimal
13650.ca2 13650bz2 \([1, 1, 1, -3497341588, 77555363485781]\) \(302773487204995438715379645049/8911747415025000000000000\) \(139246053359765625000000000000\) \([2, 2]\) \(20643840\) \(4.3694\)  
13650.ca1 13650bz3 \([1, 1, 1, -8273749588, -180064978402219]\) \(4008766897254067912673785886329/1423480510711669921875000000\) \(22241882979869842529296875000000\) \([2]\) \(41287680\) \(4.7160\)  
13650.ca4 13650bz4 \([1, 1, 1, 877658412, 258645363485781]\) \(4784981304203817469820354951/1852343836482910078035000000\) \(-28942872445045469969296875000000\) \([2]\) \(41287680\) \(4.7160\)