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Properties

Label 20280.f
Number of curves $6$
Conductor $20280$
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, -1, 0, -411278456, 3210484668156]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -411278456, 3210484668156]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -411278456, 3210484668156]) 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 20280.f have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(5\)\(1 + T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 20280.f do not have complex multiplication.

Modular form 20280.2.a.f

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);
 
\(q - q^{3} - q^{5} + q^{9} + 4 q^{11} + q^{15} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

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

Elliptic curves in class 20280.f

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
20280.f1 20280a4 \([0, -1, 0, -411278456, 3210484668156]\) \(1556580279686303289604/114075\) \(563833074355200\) \([2]\) \(2580480\) \(3.2000\)  
20280.f2 20280a5 \([0, -1, 0, -90340696, -275906686004]\) \(8248670337458940482/1446075439453125\) \(14294896529062500000000000\) \([2]\) \(5160960\) \(3.5466\)  
20280.f3 20280a3 \([0, -1, 0, -26276176, 47824146460]\) \(405929061432816484/35083409765625\) \(173405100039584400000000\) \([2, 2]\) \(2580480\) \(3.2000\)  
20280.f4 20280a2 \([0, -1, 0, -25704956, 50170032756]\) \(1520107298839022416/13013105625\) \(16079814489267360000\) \([2, 2]\) \(1290240\) \(2.8535\)  
20280.f5 20280a1 \([0, -1, 0, -1570911, 820737540]\) \(-5551350318708736/550618236675\) \(-42523664965552321200\) \([2]\) \(645120\) \(2.5069\) \(\Gamma_0(N)\)-optimal
20280.f6 20280a6 \([0, -1, 0, 28648824, 221409116460]\) \(263059523447441758/2294739983908125\) \(-22684204251110590306560000\) \([2]\) \(5160960\) \(3.5466\)