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Properties

Label 8190bz
Number of curves $6$
Conductor $8190$
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 8190bz have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(7\)\(1 - T\)
\(13\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 8190bz do not have complex multiplication.

Modular form 8190.2.a.bz

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} + q^{10} + 6 q^{11} + q^{13} + q^{14} + q^{16} - 6 q^{17} + 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 8190bz

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8190.ca4 8190bz1 \([1, -1, 1, -129092, -17820061]\) \(326355561310674169/465699780\) \(339495139620\) \([2]\) \(41472\) \(1.4840\) \(\Gamma_0(N)\)-optimal
8190.ca5 8190bz2 \([1, -1, 1, -127922, -18159829]\) \(-317562142497484249/12339342574650\) \(-8995380736919850\) \([2]\) \(82944\) \(1.8305\)  
8190.ca3 8190bz3 \([1, -1, 1, -164327, -7302049]\) \(673163386034885929/357608625192000\) \(260696687764968000\) \([6]\) \(124416\) \(2.0333\)  
8190.ca6 8190bz4 \([1, -1, 1, 626593, -57604561]\) \(37321015309599759191/23553520979625000\) \(-17170516794146625000\) \([6]\) \(248832\) \(2.3798\)  
8190.ca1 8190bz5 \([1, -1, 1, -7670462, 8178473549]\) \(68463752473882049153689/1817088000000000\) \(1324657152000000000\) \([6]\) \(373248\) \(2.5826\)  
8190.ca2 8190bz6 \([1, -1, 1, -7370942, 8846283341]\) \(-60752633741424905775769/11197265625000000000\) \(-8162806640625000000000\) \([6]\) \(746496\) \(2.9291\)