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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 0, 144032, 6591488]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, 144032, 6591488]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, 144032, 6591488]) 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 61370d have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(5\)\(1 - T\)
\(17\)\(1 + T\)
\(19\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + T + 3 T^{2}\) 1.3.b
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 + 5 T + 11 T^{2}\) 1.11.f
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(23\) \( 1 - 3 T + 23 T^{2}\) 1.23.ad
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 61370d do not have complex multiplication.

Modular form 61370.2.a.d

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);
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + 2 q^{7} - q^{8} - 2 q^{9} + q^{10} - q^{12} + q^{13} - 2 q^{14} + q^{15} + q^{16} - q^{17} + 2 q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

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 61370d

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
61370.c2 61370d1 \([1, 1, 0, 144032, 6591488]\) \(7023836099951/4456448000\) \(-209657522290688000\) \([]\) \(603288\) \(2.0127\) \(\Gamma_0(N)\)-optimal
61370.c1 61370d2 \([1, 1, 0, -2397408, 1473834112]\) \(-32391289681150609/1228250000000\) \(-57784103338250000000\) \([]\) \(1809864\) \(2.5620\)