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Properties

Label 126150.ch
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, 1, 1, -4211963, 3284118281]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -4211963, 3284118281]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -4211963, 3284118281]) 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.ch 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.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 126150.2.a.ch

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.ch

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.ch1 126150cg3 \([1, 1, 1, -4211963, 3284118281]\) \(21685195471991381/309586821120\) \(117976765317120000000\) \([2]\) \(5376000\) \(2.6558\)  
126150.ch2 126150cg4 \([1, 1, 1, -499963, 8874390281]\) \(-36267977929301/89261680665600\) \(-34015673902395600000000\) \([2]\) \(10752000\) \(3.0024\)  
126150.ch3 126150cg1 \([1, 1, 1, -423838, -106380469]\) \(22095784790981/450000\) \(171485156250000\) \([2]\) \(1075200\) \(1.8511\) \(\Gamma_0(N)\)-optimal
126150.ch4 126150cg2 \([1, 1, 1, -409338, -113978469]\) \(-19904714311301/3164062500\) \(-1205755004882812500\) \([2]\) \(2150400\) \(2.1977\)