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Properties

Label 100800fu
Number of curves $6$
Conductor $100800$
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, -57900, -11842000]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -57900, -11842000]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -57900, -11842000]) 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 100800fu have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1\)
\(7\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 2 T + 11 T^{2}\) 1.11.ac
\(13\) \( 1 - 3 T + 13 T^{2}\) 1.13.ad
\(17\) \( 1 + T + 17 T^{2}\) 1.17.b
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 9 T + 23 T^{2}\) 1.23.j
\(29\) \( 1 - 3 T + 29 T^{2}\) 1.29.ad
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 100800fu do not have complex multiplication.

Modular form 100800.2.a.fu

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);
 
\(q + q^{7} - 4 q^{11} + 6 q^{13} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

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 100800fu

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
100800.jq5 100800fu1 \([0, 0, 0, -57900, -11842000]\) \(-7189057/16128\) \(-48157949952000000\) \([2]\) \(786432\) \(1.8897\) \(\Gamma_0(N)\)-optimal
100800.jq4 100800fu2 \([0, 0, 0, -1209900, -511810000]\) \(65597103937/63504\) \(189621927936000000\) \([2, 2]\) \(1572864\) \(2.2363\)  
100800.jq3 100800fu3 \([0, 0, 0, -1497900, -249730000]\) \(124475734657/63011844\) \(188152357994496000000\) \([2, 2]\) \(3145728\) \(2.5828\)  
100800.jq1 100800fu4 \([0, 0, 0, -19353900, -32771842000]\) \(268498407453697/252\) \(752467968000000\) \([2]\) \(3145728\) \(2.5828\)  
100800.jq6 100800fu5 \([0, 0, 0, 5558100, -1929058000]\) \(6359387729183/4218578658\) \(-12596608375529472000000\) \([2]\) \(6291456\) \(2.9294\)  
100800.jq2 100800fu6 \([0, 0, 0, -13161900, 18202718000]\) \(84448510979617/933897762\) \(2788603774967808000000\) \([2]\) \(6291456\) \(2.9294\)