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Properties

Label 130050.cj
Number of curves $2$
Conductor $130050$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cj1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 130050.cj have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(5\)\(1\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - T + 7 T^{2}\) 1.7.ab
\(11\) \( 1 - 3 T + 11 T^{2}\) 1.11.ad
\(13\) \( 1 - T + 13 T^{2}\) 1.13.ab
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 - 3 T + 23 T^{2}\) 1.23.ad
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 130050.cj do not have complex multiplication.

Modular form 130050.2.a.cj

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{7} - q^{8} + 3 q^{11} + q^{13} - q^{14} + q^{16} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.

Elliptic curves in class 130050.cj

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
130050.cj1 130050fo2 \([1, -1, 0, -5365917, 4804777741]\) \(-215038729/1000\) \(-79458237101390625000\) \([]\) \(5287680\) \(2.6690\)  
130050.cj2 130050fo1 \([1, -1, 0, 161208, 34868866]\) \(5831/10\) \(-794582371013906250\) \([]\) \(1762560\) \(2.1197\) \(\Gamma_0(N)\)-optimal