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Properties

Label 167310.co
Number of curves $2$
Conductor $167310$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("co1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 167310.co have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(11\)\(1 - T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 167310.co do not have complex multiplication.

Modular form 167310.2.a.co

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} + 2 q^{7} - q^{8} - q^{10} + q^{11} - 2 q^{14} + q^{16} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 167310.co

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
167310.co1 167310dp2 \([1, -1, 0, -36842454, 86066330260]\) \(715404503176453/160380000\) \(1239846465283028460000\) \([2]\) \(14376960\) \(3.0401\)  
167310.co2 167310dp1 \([1, -1, 0, -2041974, 1661246068]\) \(-121802655493/83635200\) \(-646556971525371878400\) \([2]\) \(7188480\) \(2.6935\) \(\Gamma_0(N)\)-optimal