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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("fk1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 167310.fk 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
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 7 T + 19 T^{2}\) 1.19.h
\(23\) \( 1 - 3 T + 23 T^{2}\) 1.23.ad
\(29\) \( 1 - 3 T + 29 T^{2}\) 1.29.ad
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 167310.2.a.fk

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{5} + 2 q^{7} + q^{8} + q^{10} + q^{11} + 2 q^{14} + q^{16} - 6 q^{17} - 7 q^{19} + O(q^{20})\) 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 167310.fk

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
167310.fk1 167310q1 \([1, -1, 1, -806162, -1298657239]\) \(-97435188409/1169170200\) \(-695267748608713651800\) \([]\) \(8626176\) \(2.6811\) \(\Gamma_0(N)\)-optimal
167310.fk2 167310q2 \([1, -1, 1, 7201903, 33642131969]\) \(69469028428631/862488000000\) \(-512893751450415192000000\) \([3]\) \(25878528\) \(3.2304\)