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Properties

Label 122694dh
Number of curves $4$
Conductor $122694$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("dh1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 122694dh have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(11\)\(1\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 3 T + 5 T^{2}\) 1.5.ad
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(17\) \( 1 - 3 T + 17 T^{2}\) 1.17.ad
\(19\) \( 1 + 2 T + 19 T^{2}\) 1.19.c
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 + 3 T + 29 T^{2}\) 1.29.d
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 122694dh do not have complex multiplication.

Modular form 122694.2.a.dh

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + 4 q^{5} + q^{6} - 2 q^{7} + q^{8} + q^{9} + 4 q^{10} + q^{12} - 2 q^{14} + 4 q^{15} + q^{16} + 2 q^{17} + q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 122694dh

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
122694.dk3 122694dh1 \([1, 0, 0, -920631, -247906647]\) \(10091699281/2737152\) \(23405350016269698048\) \([2]\) \(5184000\) \(2.4246\) \(\Gamma_0(N)\)-optimal
122694.dk4 122694dh2 \([1, 0, 0, 2351209, -1612263927]\) \(168105213359/228637728\) \(-1955078143546528215072\) \([2]\) \(10368000\) \(2.7711\)  
122694.dk1 122694dh3 \([1, 0, 0, -205819611, 1136505868653]\) \(112763292123580561/1932612\) \(16525739274122523588\) \([2]\) \(25920000\) \(3.2293\)  
122694.dk2 122694dh4 \([1, 0, 0, -205615121, 1138876930203]\) \(-112427521449300721/466873642818\) \(-3992230253755059819556482\) \([2]\) \(51840000\) \(3.5758\)