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Properties

Label 262080na
Number of curves $6$
Conductor $262080$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 0, -216012, 38616464]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -216012, 38616464]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -216012, 38616464]) 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 262080na have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(7\)\(1 - T\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 262080na do not have complex multiplication.

Modular form 262080.2.a.na

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);
 
\(q + q^{5} + q^{7} + 4 q^{11} - q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 262080na

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
262080.na5 262080na1 \([0, 0, 0, -216012, 38616464]\) \(5832972054001/4542720\) \(868127311134720\) \([2]\) \(1572864\) \(1.7977\) \(\Gamma_0(N)\)-optimal
262080.na4 262080na2 \([0, 0, 0, -262092, 20940176]\) \(10418796526321/5038160400\) \(962807446005350400\) \([2, 2]\) \(3145728\) \(2.1443\)  
262080.na6 262080na3 \([0, 0, 0, 947508, 159802256]\) \(492271755328079/342606902820\) \(-65473198727044792320\) \([2]\) \(6291456\) \(2.4909\)  
262080.na2 262080na4 \([0, 0, 0, -2208972, -1249204336]\) \(6237734630203441/82168222500\) \(15702591852380160000\) \([2, 2]\) \(6291456\) \(2.4909\)  
262080.na3 262080na5 \([0, 0, 0, -336972, -3298669936]\) \(-22143063655441/24584858584650\) \(-4698239640065762918400\) \([2]\) \(12582912\) \(2.8375\)  
262080.na1 262080na6 \([0, 0, 0, -35231052, -80488987504]\) \(25306558948218234961/4478906250\) \(855932313600000000\) \([2]\) \(12582912\) \(2.8375\)