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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ff1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 40950.ff 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 \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 40950.2.a.ff

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

Elliptic curves in class 40950.ff

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
40950.ff1 40950ef6 \([1, -1, 1, -13762130, 19654072247]\) \(25306558948218234961/4478906250\) \(51017541503906250\) \([2]\) \(1572864\) \(2.6025\)  
40950.ff2 40950ef4 \([1, -1, 1, -862880, 305197247]\) \(6237734630203441/82168222500\) \(935947409414062500\) \([2, 2]\) \(786432\) \(2.2559\)  
40950.ff3 40950ef5 \([1, -1, 1, -131630, 805372247]\) \(-22143063655441/24584858584650\) \(-280036904815778906250\) \([2]\) \(1572864\) \(2.6025\)  
40950.ff4 40950ef2 \([1, -1, 1, -102380, -5086753]\) \(10418796526321/5038160400\) \(57387795806250000\) \([2, 2]\) \(393216\) \(1.9093\)  
40950.ff5 40950ef1 \([1, -1, 1, -84380, -9406753]\) \(5832972054001/4542720\) \(51744420000000\) \([2]\) \(196608\) \(1.5627\) \(\Gamma_0(N)\)-optimal
40950.ff6 40950ef3 \([1, -1, 1, 370120, -39106753]\) \(492271755328079/342606902820\) \(-3902506752434062500\) \([2]\) \(786432\) \(2.2559\)