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Properties

Label 126150n
Number of curves $4$
Conductor $126150$
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([1, 1, 0, -2540, -68100]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -2540, -68100]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -2540, -68100]) 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 126150n have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(5\)\(1\)
\(29\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 3 T + 17 T^{2}\) 1.17.ad
\(19\) \( 1 + T + 19 T^{2}\) 1.19.b
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 126150n do not have complex multiplication.

Modular form 126150.2.a.n

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);
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} - 2 q^{7} - q^{8} + q^{9} - 2 q^{11} - q^{12} - 6 q^{13} + 2 q^{14} + q^{16} + 2 q^{17} - q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 126150n

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
126150.c4 126150n1 \([1, 1, 0, -2540, -68100]\) \(-24389/12\) \(-892234981500\) \([2]\) \(201600\) \(0.99838\) \(\Gamma_0(N)\)-optimal
126150.c2 126150n2 \([1, 1, 0, -44590, -3642350]\) \(131872229/18\) \(1338352472250\) \([2]\) \(403200\) \(1.3449\)  
126150.c3 126150n3 \([1, 1, 0, -23565, 6680925]\) \(-19465109/248832\) \(-18501384576384000\) \([2]\) \(1008000\) \(1.8031\)  
126150.c1 126150n4 \([1, 1, 0, -696365, 222649725]\) \(502270291349/1889568\) \(140494889126916000\) \([2]\) \(2016000\) \(2.1497\)