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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("mi1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 286650mi 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 \(\mathbb{F}_p\)
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 286650.2.a.mi

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

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
286650.mi3 286650mi1 \([1, -1, 1, -3510757130, -63610273016503]\) \(3571003510905229697089/762141946675200000\) \(1021343100276886732800000000000\) \([2]\) \(424673280\) \(4.4726\) \(\Gamma_0(N)\)-optimal
286650.mi2 286650mi2 \([1, -1, 1, -17961445130, 870886818567497]\) \(478202393398338853167169/32244226560000000000\) \(43210347448380840000000000000000\) \([2, 2]\) \(849346560\) \(4.8191\)  
286650.mi1 286650mi3 \([1, -1, 1, -282561445130, 57811748418567497]\) \(1861772567578966373029167169/9401133413380800000\) \(12598417904205636123075000000000\) \([2]\) \(1698693120\) \(5.1657\)  
286650.mi4 286650mi4 \([1, -1, 1, 15427546870, 3737665671687497]\) \(303025056761573589385151/4678857421875000000000\) \(-6270116434160614013671875000000000\) \([2]\) \(1698693120\) \(5.1657\)