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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, 1992, -57488]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, 1992, -57488]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, 1992, -57488]) 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 11600t have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(5\)\(1\)
\(29\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - T + 3 T^{2}\) 1.3.ab
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(11\) \( 1 + 3 T + 11 T^{2}\) 1.11.d
\(13\) \( 1 + 5 T + 13 T^{2}\) 1.13.f
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 11600t do not have complex multiplication.

Modular form 11600.2.a.t

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);
 
\(q - q^{3} - 2 q^{7} - 2 q^{9} + 3 q^{11} + q^{13} - 8 q^{17} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 11600t

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
11600.g2 11600t1 \([0, -1, 0, 1992, -57488]\) \(13651919/29696\) \(-1900544000000\) \([]\) \(13440\) \(1.0403\) \(\Gamma_0(N)\)-optimal
11600.g1 11600t2 \([0, -1, 0, -182008, 30150512]\) \(-10418796526321/82044596\) \(-5250854144000000\) \([]\) \(67200\) \(1.8450\)