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Properties

Label 358160.cg
Number of curves $4$
Conductor $358160$
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, -1, 0, -10212440, -12558119440]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -10212440, -12558119440]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -10212440, -12558119440]) 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 358160.cg have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(5\)\(1 - T\)
\(11\)\(1\)
\(37\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - 2 T + 3 T^{2}\) 1.3.ac
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 358160.cg do not have complex multiplication.

Modular form 358160.2.a.cg

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);
 
\(q + 2 q^{3} + q^{5} + 2 q^{7} + q^{9} - 2 q^{13} + 2 q^{15} - 6 q^{17} + 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

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 358160.cg

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
358160.cg1 358160cg3 \([0, -1, 0, -10212440, -12558119440]\) \(16232905099479601/4052240\) \(29404325259837440\) \([2]\) \(9953280\) \(2.5356\)  
358160.cg2 358160cg4 \([0, -1, 0, -10173720, -12658109968]\) \(-16048965315233521/256572640900\) \(-1861771609233182310400\) \([2]\) \(19906560\) \(2.8822\)  
358160.cg3 358160cg1 \([0, -1, 0, -145240, -11600400]\) \(46694890801/18944000\) \(137463609688064000\) \([2]\) \(3317760\) \(1.9863\) \(\Gamma_0(N)\)-optimal
358160.cg4 358160cg2 \([0, -1, 0, 474280, -84951568]\) \(1625964918479/1369000000\) \(-9933893668864000000\) \([2]\) \(6635520\) \(2.3329\)