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Properties

Label 22848.bc
Number of curves $6$
Conductor $22848$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bc1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 22848.bc have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(7\)\(1 + T\)
\(17\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 22848.bc do not have complex multiplication.

Modular form 22848.2.a.bc

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + 2 q^{5} - q^{7} + q^{9} + 4 q^{11} + 2 q^{13} - 2 q^{15} + q^{17} + 4 q^{19} + O(q^{20})\) 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 22848.bc

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
22848.bc1 22848bx6 \([0, -1, 0, -877990657, 10013724179617]\) \(285531136548675601769470657/17941034271597192\) \(4703134488093574299648\) \([4]\) \(5898240\) \(3.6189\)  
22848.bc2 22848bx4 \([0, -1, 0, -54978817, 155852962465]\) \(70108386184777836280897/552468975892674624\) \(144826427216409296633856\) \([2, 2]\) \(2949120\) \(3.2723\)  
22848.bc3 22848bx5 \([0, -1, 0, -18726657, 358277773473]\) \(-2770540998624539614657/209924951154647363208\) \(-55030566395483878380797952\) \([2]\) \(5898240\) \(3.6189\)  
22848.bc4 22848bx2 \([0, -1, 0, -5806337, -1351456095]\) \(82582985847542515777/44772582831427584\) \(11736863953761752580096\) \([2, 2]\) \(1474560\) \(2.9257\)  
22848.bc5 22848bx1 \([0, -1, 0, -4495617, -3662779743]\) \(38331145780597164097/55468445663232\) \(14540720219942289408\) \([2]\) \(737280\) \(2.5791\) \(\Gamma_0(N)\)-optimal
22848.bc6 22848bx3 \([0, -1, 0, 22394623, -10652132703]\) \(4738217997934888496063/2928751705237796928\) \(-767754687017857037893632\) \([2]\) \(2949120\) \(3.2723\)