↑

Properties

Label 286650bm
Number of curves $4$
Conductor $286650$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("bm1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 286650bm have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(5\)\(1\)
\(7\)\(1\)
\(13\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 8 T + 19 T^{2}\) 1.19.ai
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 286650bm do not have complex multiplication.

Modular form 286650.2.a.bm

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{8} - 4 q^{11} + q^{13} + q^{16} + 6 q^{17} + 8 q^{19} + O(q^{20})\) 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 286650bm

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
286650.bm4 286650bm1 \([1, -1, 0, -7938479817, 137915887955341]\) \(41285728533151645510969/17760741842188800000\) \(23801092716983242588200000000000\) \([2]\) \(1061683200\) \(4.7170\) \(\Gamma_0(N)\)-optimal
286650.bm2 286650bm2 \([1, -1, 0, -108701687817, 13788609965339341]\) \(105997782562506306791694649/51649016225625000000\) \(69214621486529953916015625000000\) \([2, 2]\) \(2123366400\) \(5.0636\)  
286650.bm1 286650bm3 \([1, -1, 0, -1739023562817, 882686586787214341]\) \(434014578033107719741685694649/103121648659575000\) \(138192871822759332165234375000\) \([2]\) \(4246732800\) \(5.4101\)  
286650.bm3 286650bm4 \([1, -1, 0, -90591140817, 18533844937544341]\) \(-61354313914516350666047929/75227254486083984375000\) \(-100811715792988622188568115234375000\) \([2]\) \(4246732800\) \(5.4101\)