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Properties

Label 60552.u
Number of curves $6$
Conductor $60552$
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, -2909019, 1909707478]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -2909019, 1909707478]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -2909019, 1909707478]) 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 60552.u have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(29\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 60552.u do not have complex multiplication.

Modular form 60552.2.a.u

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

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

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 60552.u

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
60552.u1 60552q6 \([0, 0, 0, -2909019, 1909707478]\) \(3065617154/9\) \(7992598136997888\) \([2]\) \(774144\) \(2.2808\)  
60552.u2 60552q4 \([0, 0, 0, -486939, -130773818]\) \(28756228/3\) \(1332099689499648\) \([2]\) \(387072\) \(1.9342\)  
60552.u3 60552q3 \([0, 0, 0, -184179, 29022910]\) \(1556068/81\) \(35966691616490496\) \([2, 2]\) \(387072\) \(1.9342\)  
60552.u4 60552q2 \([0, 0, 0, -32799, -1707230]\) \(35152/9\) \(999074767124736\) \([2, 2]\) \(193536\) \(1.5876\)  
60552.u5 60552q1 \([0, 0, 0, 5046, -170723]\) \(2048/3\) \(-20814057648432\) \([2]\) \(96768\) \(1.2410\) \(\Gamma_0(N)\)-optimal
60552.u6 60552q5 \([0, 0, 0, 118581, 115067302]\) \(207646/6561\) \(-5826604041871460352\) \([2]\) \(774144\) \(2.2808\)