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Properties

Label 330096f
Number of curves $4$
Conductor $330096$
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, -165911504, 834089722368]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -165911504, 834089722368]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -165911504, 834089722368]) 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 330096f have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(13\)\(1 - T\)
\(23\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 8 T + 19 T^{2}\) 1.19.i
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 330096f do not have complex multiplication.

Modular form 330096.2.a.f

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);
 
\(q - q^{3} - 2 q^{5} - 4 q^{7} + q^{9} - 4 q^{11} + q^{13} + 2 q^{15} - 6 q^{17} - 8 q^{19} + O(q^{20})\) 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.

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

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 330096f

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
330096.f4 330096f1 \([0, -1, 0, -165911504, 834089722368]\) \(-832964037319114273/13553130966687\) \(-8218008745520903904227328\) \([2]\) \(113541120\) \(3.5809\) \(\Gamma_0(N)\)-optimal
330096.f3 330096f2 \([0, -1, 0, -2664865184, 52950268509504]\) \(3451616026781746657393/892157700681\) \(540964283795487704715264\) \([2, 2]\) \(227082240\) \(3.9275\)  
330096.f1 330096f3 \([0, -1, 0, -42637840304, 3388774988223744]\) \(14137816614617731097417473/944541\) \(572727159324463104\) \([4]\) \(454164480\) \(4.2740\)  
330096.f2 330096f4 \([0, -1, 0, -2675148944, 52521007913088]\) \(3491729964247447364833/55470800868972723\) \(33634997535458711822564339712\) \([2]\) \(454164480\) \(4.2740\)