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Properties

Label 117648.bg
Number of curves $4$
Conductor $117648$
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, -3942075, -2738410774]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -3942075, -2738410774]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -3942075, -2738410774]) 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 117648.bg have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(19\)\(1 + T\)
\(43\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(23\) \( 1 + 23 T^{2}\) 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 117648.bg do not have complex multiplication.

Modular form 117648.2.a.bg

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

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

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 117648.bg

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
117648.bg1 117648x3 \([0, 0, 0, -3942075, -2738410774]\) \(2268876641163765625/228097945239552\) \(681096814918178439168\) \([2]\) \(3981312\) \(2.7337\)  
117648.bg2 117648x1 \([0, 0, 0, -864075, 308538362]\) \(23894093340015625/55042322688\) \(164355494869204992\) \([2]\) \(1327104\) \(2.1844\) \(\Gamma_0(N)\)-optimal
117648.bg3 117648x2 \([0, 0, 0, -553035, 533669114]\) \(-6264610702863625/37578744274608\) \(-112209529144071094272\) \([2]\) \(2654208\) \(2.5310\)  
117648.bg4 117648x4 \([0, 0, 0, 4905285, -13279155478]\) \(4371484788393482375/28041364201746432\) \(-83731064844587618009088\) \([2]\) \(7962624\) \(3.0803\)