↑

Properties

Label 122304in
Number of curves $6$
Conductor $122304$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 1, 0, 141251, -83128333]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, 141251, -83128333]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, 141251, -83128333]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 122304in have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(7\)\(1\)
\(13\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 3 T + 5 T^{2}\) 1.5.ad
\(11\) \( 1 - 2 T + 11 T^{2}\) 1.11.ac
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 - 7 T + 19 T^{2}\) 1.19.ah
\(23\) \( 1 + T + 23 T^{2}\) 1.23.b
\(29\) \( 1 - 3 T + 29 T^{2}\) 1.29.ad
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 122304in do not have complex multiplication.

Modular form 122304.2.a.in

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q + q^{3} - 2 q^{5} + q^{9} - 4 q^{11} + q^{13} - 2 q^{15} + 6 q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 122304in

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
122304.ff6 122304in1 \([0, 1, 0, 141251, -83128333]\) \(2587063175168/26304786963\) \(-3169005446563826688\) \([2]\) \(1769472\) \(2.2311\) \(\Gamma_0(N)\)-optimal
122304.ff5 122304in2 \([0, 1, 0, -2211729, -1175381649]\) \(620742479063632/49991146569\) \(96360995269775867904\) \([2, 2]\) \(3538944\) \(2.5777\)  
122304.ff4 122304in3 \([0, 1, 0, -7397889, 6378779007]\) \(5807363790481348/1079211743883\) \(8320986805442384166912\) \([2]\) \(7077888\) \(2.9242\)  
122304.ff2 122304in4 \([0, 1, 0, -34673249, -78596106849]\) \(597914615076708388/4400862921\) \(33931730733808287744\) \([2, 2]\) \(7077888\) \(2.9242\)  
122304.ff3 122304in5 \([0, 1, 0, -33959809, -81984518785]\) \(-280880296871140514/25701087819771\) \(-396323360723204620812288\) \([2]\) \(14155776\) \(3.2708\)  
122304.ff1 122304in6 \([0, 1, 0, -554771009, -5029614723393]\) \(1224522642327678150914/66339\) \(1022979868065792\) \([2]\) \(14155776\) \(3.2708\)