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Properties

Label 167310eh
Number of curves $4$
Conductor $167310$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("eh1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 167310eh 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 \(\mathbb{F}_p\)
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 6 T + 19 T^{2}\) 1.19.ag
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 4 T + 29 T^{2}\) 1.29.ae
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 167310.2.a.eh

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{5} - q^{8} + q^{10} + q^{11} + q^{16} - 6 q^{17} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 167310eh

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
167310.q4 167310eh1 \([1, -1, 0, -2661093720, -19347322483904]\) \(592265697637387401314569/296787655248366796800\) \(1044319710247009571679554764800\) \([2]\) \(227082240\) \(4.4528\) \(\Gamma_0(N)\)-optimal
167310.q2 167310eh2 \([1, -1, 0, -34558775640, -2470805389227200]\) \(1297212465095901089487274249/1193746061037404160000\) \(4200486504493691075635445760000\) \([2, 2]\) \(454164480\) \(4.7994\)  
167310.q3 167310eh3 \([1, -1, 0, -26662230360, -3629921898478784]\) \(-595697118196750093952139529/1272946549598037600000000\) \(-4479172729484571862843413600000000\) \([2]\) \(908328960\) \(5.1460\)  
167310.q1 167310eh4 \([1, -1, 0, -552818231640, -158205387923729600]\) \(5309860874757074224246393258249/4502770931800627200\) \(15844097123485613455886899200\) \([2]\) \(908328960\) \(5.1460\)