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Properties

Label 227136hq
Number of curves $4$
Conductor $227136$
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, -28929, -1881855]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -28929, -1881855]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -28929, -1881855]) 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 227136hq have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(7\)\(1 + T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - T + 5 T^{2}\) 1.5.ab
\(11\) \( 1 - 2 T + 11 T^{2}\) 1.11.ac
\(17\) \( 1 + 3 T + 17 T^{2}\) 1.17.d
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 9 T + 29 T^{2}\) 1.29.j
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 227136hq do not have complex multiplication.

Modular form 227136.2.a.hq

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);
 
\(q - q^{3} - 2 q^{5} - q^{7} + q^{9} + 2 q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 227136hq

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
227136.v3 227136hq1 \([0, -1, 0, -28929, -1881855]\) \(4649101309/6804\) \(3918630223872\) \([2]\) \(460800\) \(1.3180\) \(\Gamma_0(N)\)-optimal
227136.v4 227136hq2 \([0, -1, 0, -20609, -2995071]\) \(-1680914269/5786802\) \(-3332795005403136\) \([2]\) \(921600\) \(1.6646\)  
227136.v1 227136hq3 \([0, -1, 0, -865089, 309876033]\) \(124318741396429/51631104\) \(29735920726966272\) \([2]\) \(2304000\) \(2.1228\)  
227136.v2 227136hq4 \([0, -1, 0, -731969, 408358209]\) \(-75306487574989/81352871712\) \(-46853589342948950016\) \([2]\) \(4608000\) \(2.4693\)