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Properties

Label 126150.bk
Number of curves $4$
Conductor $126150$
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([1, 0, 1, -3542260901, 80138867890448]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, -3542260901, 80138867890448]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, -3542260901, 80138867890448]) 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 126150.bk have rank \(1\).

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
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 6 T + 13 T^{2}\) 1.13.g
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 126150.2.a.bk

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);
 
\(q - q^{2} + q^{3} + q^{4} - q^{6} + 2 q^{7} - q^{8} + q^{9} + q^{12} - 6 q^{13} - 2 q^{14} + q^{16} + 2 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

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

Elliptic curves in class 126150.bk

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.bk1 126150bi3 \([1, 0, 1, -3542260901, 80138867890448]\) \(21685195471991381/309586821120\) \(70175331346766936555520000000\) \([2]\) \(155904000\) \(4.3395\)  
126150.bk2 126150bi4 \([1, 0, 1, -420468901, 216442550194448]\) \(-36267977929301/89261680665600\) \(-20233316116675980647787600000000\) \([2]\) \(311808000\) \(4.6860\)  
126150.bk3 126150bi1 \([1, 0, 1, -356447776, -2590235880802]\) \(22095784790981/450000\) \(102003370142828906250000\) \([2]\) \(31180800\) \(3.5347\) \(\Gamma_0(N)\)-optimal
126150.bk4 126150bi2 \([1, 0, 1, -344253276, -2775689836802]\) \(-19904714311301/3164062500\) \(-717211196316765747070312500\) \([2]\) \(62361600\) \(3.8813\)