Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, -60, -184]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -60, -184]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -60, -184]) 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 curve 782.d1 has rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(17\)\(1 + T\)
\(23\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - T + 3 T^{2}\) 1.3.ab
\(5\) \( 1 + 4 T + 5 T^{2}\) 1.5.e
\(7\) \( 1 - 3 T + 7 T^{2}\) 1.7.ad
\(11\) \( 1 - 6 T + 11 T^{2}\) 1.11.ag
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(19\) \( 1 + 3 T + 19 T^{2}\) 1.19.d
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 782.d do not have complex multiplication.

Modular form 782.2.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);
 
\(q + q^{2} + q^{3} + q^{4} - 4 q^{5} + q^{6} + 3 q^{7} + q^{8} - 2 q^{9} - 4 q^{10} + 6 q^{11} + q^{12} + 6 q^{13} + 3 q^{14} - 4 q^{15} + q^{16} - q^{17} - 2 q^{18} - 3 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny graph

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

Elliptic curves in class 782.d

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
782.d1 782b1 \([1, 0, 0, -60, -184]\) \(-23912763841/3128\) \(-3128\) \([]\) \(168\) \(-0.30903\) \(\Gamma_0(N)\)-optimal