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Properties

Label 40950z
Number of curves $6$
Conductor $40950$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("z1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 40950z have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 40950z do not have complex multiplication.

Modular form 40950.2.a.z

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{7} - q^{8} + 6 q^{11} - q^{13} + q^{14} + q^{16} + 6 q^{17} + 2 q^{19} + O(q^{20})\) 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 40950z

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
40950.bm4 40950z1 \([1, -1, 0, -3227292, -2230734884]\) \(326355561310674169/465699780\) \(5304611556562500\) \([2]\) \(995328\) \(2.2887\) \(\Gamma_0(N)\)-optimal
40950.bm5 40950z2 \([1, -1, 0, -3198042, -2273176634]\) \(-317562142497484249/12339342574650\) \(-140552824014372656250\) \([2]\) \(1990656\) \(2.6353\)  
40950.bm3 40950z3 \([1, -1, 0, -4108167, -916864259]\) \(673163386034885929/357608625192000\) \(4073385746327625000000\) \([2]\) \(2985984\) \(2.8380\)  
40950.bm6 40950z4 \([1, -1, 0, 15664833, -7184905259]\) \(37321015309599759191/23553520979625000\) \(-268289324908541015625000\) \([2]\) \(5971968\) \(3.1846\)  
40950.bm1 40950z5 \([1, -1, 0, -191761542, 1022117432116]\) \(68463752473882049153689/1817088000000000\) \(20697768000000000000000\) \([2]\) \(8957952\) \(3.3873\)  
40950.bm2 40950z6 \([1, -1, 0, -184273542, 1105601144116]\) \(-60752633741424905775769/11197265625000000000\) \(-127543853759765625000000000\) \([2]\) \(17915904\) \(3.7339\)