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Properties

Label 26208q
Number of curves $4$
Conductor $26208$
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, -82281, -9029680]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -82281, -9029680]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -82281, -9029680]) 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 26208q have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(7\)\(1 - T\)
\(13\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(17\) \( 1 - 4 T + 17 T^{2}\) 1.17.ae
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 26208q do not have complex multiplication.

Modular form 26208.2.a.q

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);
 
\(q - 2 q^{5} + q^{7} - q^{13} + 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

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 26208q

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
26208.p3 26208q1 \([0, 0, 0, -82281, -9029680]\) \(1320428512222912/9182047329\) \(428397600181824\) \([2, 2]\) \(98304\) \(1.6409\) \(\Gamma_0(N)\)-optimal
26208.p4 26208q2 \([0, 0, 0, -31251, -20103190]\) \(-9043113453704/462519318807\) \(-172634410706075136\) \([2]\) \(196608\) \(1.9874\)  
26208.p2 26208q3 \([0, 0, 0, -135516, 4087424]\) \(92173898928448/50924270943\) \(152059058247462912\) \([2]\) \(196608\) \(1.9874\)  
26208.p1 26208q4 \([0, 0, 0, -1314291, -579943114]\) \(672668087746709384/32867289\) \(12267649884672\) \([2]\) \(196608\) \(1.9874\)