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Properties

Label 92736co
Number of curves $6$
Conductor $92736$
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, 0, 0, 72564, -15409424]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, 72564, -15409424]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, 72564, -15409424]) 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 92736co have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(7\)\(1 + T\)
\(23\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 + T + 11 T^{2}\) 1.11.b
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 5 T + 19 T^{2}\) 1.19.af
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 92736co do not have complex multiplication.

Modular form 92736.2.a.co

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);
 
\(q - 2 q^{5} + q^{7} - 4 q^{11} + 2 q^{13} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

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 92736co

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
92736.bk5 92736co1 \([0, 0, 0, 72564, -15409424]\) \(221115865823/664731648\) \(-127032196174184448\) \([2]\) \(786432\) \(1.9653\) \(\Gamma_0(N)\)-optimal
92736.bk4 92736co2 \([0, 0, 0, -664716, -178495760]\) \(169967019783457/26337394944\) \(5033154553885753344\) \([2, 2]\) \(1572864\) \(2.3119\)  
92736.bk3 92736co3 \([0, 0, 0, -2922636, 1749767920]\) \(14447092394873377/1439452851984\) \(275083723825829904384\) \([2, 2]\) \(3145728\) \(2.6584\)  
92736.bk2 92736co4 \([0, 0, 0, -10203276, -12544284944]\) \(614716917569296417/19093020912\) \(3648733117113434112\) \([2]\) \(3145728\) \(2.6584\)  
92736.bk6 92736co5 \([0, 0, 0, 3609204, 8461886704]\) \(27207619911317663/177609314617308\) \(-33941668588687859908608\) \([2]\) \(6291456\) \(3.0050\)  
92736.bk1 92736co6 \([0, 0, 0, -45581196, 118446524656]\) \(54804145548726848737/637608031452\) \(121848792331978801152\) \([2]\) \(6291456\) \(3.0050\)