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Properties

Label 117600.er
Number of curves $4$
Conductor $117600$
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, -396920008, 3043572972488]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -396920008, 3043572972488]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -396920008, 3043572972488]) 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 117600.er have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 117600.er do not have complex multiplication.

Modular form 117600.2.a.er

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

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

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 117600.er

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
117600.er1 117600df4 \([0, 1, 0, -396920008, 3043572972488]\) \(7347751505995469192/72930375\) \(68641485507000000000\) \([2]\) \(17694720\) \(3.3826\)  
117600.er2 117600df3 \([0, 1, 0, -35545008, 2476784988]\) \(5276930158229192/3050936350875\) \(2871516885952743000000000\) \([2]\) \(17694720\) \(3.3826\)  
117600.er3 117600df1 \([0, 1, 0, -24826258, 47474097488]\) \(14383655824793536/45209390625\) \(5318839597640625000000\) \([2, 2]\) \(8847360\) \(3.0360\) \(\Gamma_0(N)\)-optimal
117600.er4 117600df2 \([0, 1, 0, -14407633, 87637896863]\) \(-43927191786304/415283203125\) \(-3126889828125000000000000\) \([2]\) \(17694720\) \(3.3826\)