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Properties

Label 98736bz
Number of curves $6$
Conductor $98736$
CM no
Rank $1$
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 98736bz have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(11\)\(1\)
\(17\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(7\) \( 1 - 3 T + 7 T^{2}\) 1.7.ad
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(19\) \( 1 - 5 T + 19 T^{2}\) 1.19.af
\(23\) \( 1 + 4 T + 23 T^{2}\) 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 98736bz do not have complex multiplication.

Modular form 98736.2.a.bz

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} + q^{9} + 2 q^{13} + 2 q^{15} - q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 98736bz

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
98736.n5 98736bz1 \([0, -1, 0, -65864, 6055920]\) \(4354703137/352512\) \(2557937710006272\) \([2]\) \(491520\) \(1.6995\) \(\Gamma_0(N)\)-optimal
98736.n4 98736bz2 \([0, -1, 0, -220744, -32849936]\) \(163936758817/30338064\) \(220142514167414784\) \([2, 2]\) \(983040\) \(2.0460\)  
98736.n6 98736bz3 \([0, -1, 0, 437496, -191880720]\) \(1276229915423/2927177028\) \(-21240515227241299968\) \([2]\) \(1966080\) \(2.3926\)  
98736.n2 98736bz4 \([0, -1, 0, -3357064, -2366272016]\) \(576615941610337/27060804\) \(196361687019700224\) \([2, 2]\) \(1966080\) \(2.3926\)  
98736.n3 98736bz5 \([0, -1, 0, -3182824, -2623032080]\) \(-491411892194497/125563633938\) \(-911129136754021244928\) \([2]\) \(3932160\) \(2.7392\)  
98736.n1 98736bz6 \([0, -1, 0, -53712424, -151498706192]\) \(2361739090258884097/5202\) \(37747344678912\) \([2]\) \(3932160\) \(2.7392\)