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Properties

Label 141120ik
Number of curves $4$
Conductor $141120$
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, 0, 0, -1296687, -2366352884]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -1296687, -2366352884]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -1296687, -2366352884]) 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 141120ik have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 6 T + 11 T^{2}\) 1.11.g
\(13\) \( 1 - 3 T + 13 T^{2}\) 1.13.ad
\(17\) \( 1 - 4 T + 17 T^{2}\) 1.17.ae
\(19\) \( 1 + T + 19 T^{2}\) 1.19.b
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 + 8 T + 29 T^{2}\) 1.29.i
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 141120ik do not have complex multiplication.

Modular form 141120.2.a.ik

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);
 
\(q + q^{5} - 4 q^{11} - 2 q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 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 141120ik

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
141120.iz4 141120ik1 \([0, 0, 0, -1296687, -2366352884]\) \(-43927191786304/415283203125\) \(-2279502684703125000000\) \([2]\) \(5898240\) \(2.7806\) \(\Gamma_0(N)\)-optimal
141120.iz3 141120ik2 \([0, 0, 0, -35749812, -82049540384]\) \(14383655824793536/45209390625\) \(15881969937121344000000\) \([2, 2]\) \(11796480\) \(3.1271\)  
141120.iz2 141120ik3 \([0, 0, 0, -51184812, -4300358384]\) \(5276930158229192/3050936350875\) \(8574303477184715354112000\) \([2]\) \(23592960\) \(3.4737\)  
141120.iz1 141120ik4 \([0, 0, 0, -571564812, -5259522722384]\) \(7347751505995469192/72930375\) \(204962377460133888000\) \([2]\) \(23592960\) \(3.4737\)