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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 0, -12096, -509036]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -12096, -509036]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -12096, -509036]) 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 126.a have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(7\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 126.a do not have complex multiplication.

Modular form 126.2.a.a

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);
 
\(q - q^{2} + q^{4} + 2 q^{5} - q^{7} - q^{8} - 2 q^{10} + 4 q^{11} + 6 q^{13} + q^{14} + q^{16} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

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 126.a

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
126.a1 126b3 \([1, -1, 0, -12096, -509036]\) \(268498407453697/252\) \(183708\) \([2]\) \(128\) \(0.73841\)  
126.a2 126b5 \([1, -1, 0, -8226, 286474]\) \(84448510979617/933897762\) \(680811468498\) \([2]\) \(256\) \(1.0850\)  
126.a3 126b4 \([1, -1, 0, -936, -3668]\) \(124475734657/63011844\) \(45935634276\) \([2, 2]\) \(128\) \(0.73841\)  
126.a4 126b2 \([1, -1, 0, -756, -7808]\) \(65597103937/63504\) \(46294416\) \([2, 2]\) \(64\) \(0.39183\)  
126.a5 126b1 \([1, -1, 0, -36, -176]\) \(-7189057/16128\) \(-11757312\) \([2]\) \(32\) \(0.045260\) \(\Gamma_0(N)\)-optimal
126.a6 126b6 \([1, -1, 0, 3474, -31010]\) \(6359387729183/4218578658\) \(-3075343841682\) \([2]\) \(256\) \(1.0850\)