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Properties

Label 109200.dl
Number of curves $6$
Conductor $109200$
CM no
Rank $0$
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, -70761608, 229133909712]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -70761608, 229133909712]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -70761608, 229133909712]) 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 109200.dl have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(5\)\(1\)
\(7\)\(1 - T\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 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 109200.dl do not have complex multiplication.

Modular form 109200.2.a.dl

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);
 
\(q - q^{3} + q^{7} + q^{9} + 4 q^{11} - q^{13} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

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 109200.dl

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
109200.dl1 109200q6 \([0, -1, 0, -70761608, 229133909712]\) \(1224522642327678150914/66339\) \(2122848000000\) \([2]\) \(4718592\) \(2.7560\)  
109200.dl2 109200q4 \([0, -1, 0, -4422608, 3581309712]\) \(597914615076708388/4400862921\) \(70413806736000000\) \([2, 2]\) \(2359296\) \(2.4094\)  
109200.dl3 109200q5 \([0, -1, 0, -4331608, 3735645712]\) \(-280880296871140514/25701087819771\) \(-822434810232672000000\) \([2]\) \(4718592\) \(2.7560\)  
109200.dl4 109200q3 \([0, -1, 0, -943608, -290376288]\) \(5807363790481348/1079211743883\) \(17267387902128000000\) \([2]\) \(2359296\) \(2.4094\)  
109200.dl5 109200q2 \([0, -1, 0, -282108, 53603712]\) \(620742479063632/49991146569\) \(199964586276000000\) \([2, 2]\) \(1179648\) \(2.0629\)  
109200.dl6 109200q1 \([0, -1, 0, 18017, 3782962]\) \(2587063175168/26304786963\) \(-6576196740750000\) \([2]\) \(589824\) \(1.7163\) \(\Gamma_0(N)\)-optimal