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Properties

Label 124950.cb
Number of curves $6$
Conductor $124950$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cb1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 124950.cb have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 124950.cb do not have complex multiplication.

Modular form 124950.2.a.cb

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} + 4 q^{11} - q^{12} - 2 q^{13} + q^{16} + q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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 124950.cb

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
124950.cb1 124950ba6 \([1, 1, 0, -16805289925, 838520099849125]\) \(285531136548675601769470657/17941034271597192\) \(32980386578424031900125000\) \([2]\) \(188743680\) \(4.3568\)  
124950.cb2 124950ba4 \([1, 1, 0, -1052328925, 13049190488125]\) \(70108386184777836280897/552468975892674624\) \(1015584727262457450609000000\) \([2, 2]\) \(94371840\) \(4.0102\)  
124950.cb3 124950ba5 \([1, 1, 0, -358439925, 30001592647125]\) \(-2770540998624539614657/209924951154647363208\) \(-385897821537392306782156125000\) \([2]\) \(188743680\) \(4.3568\)  
124950.cb4 124950ba2 \([1, 1, 0, -111136925, -113379631875]\) \(82582985847542515777/44772582831427584\) \(82303899961478497344000000\) \([2, 2]\) \(47185920\) \(3.6637\)  
124950.cb5 124950ba1 \([1, 1, 0, -86048925, -306883375875]\) \(38331145780597164097/55468445663232\) \(101965736934899712000000\) \([2]\) \(23592960\) \(3.3171\) \(\Gamma_0(N)\)-optimal
124950.cb6 124950ba3 \([1, 1, 0, 428647075, -891208375875]\) \(4738217997934888496063/2928751705237796928\) \(-5383823583898774543473000000\) \([2]\) \(94371840\) \(4.0102\)