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Properties

Label 277440gz
Number of curves $4$
Conductor $277440$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("gz1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 277440gz have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1 + T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 277440gz do not have complex multiplication.

Modular form 277440.2.a.gz

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} - q^{5} + 4 q^{7} + q^{9} - 2 q^{13} - q^{15} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content sage:E.isogeny_class().matrix()
 

The \(i,j\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 277440gz

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
277440.gz4 277440gz1 \([0, 1, 0, -23393201, 1321974751215]\) \(-3579968623693264/1906997690433375\) \(-754160244091719335368704000\) \([2]\) \(123863040\) \(3.8364\) \(\Gamma_0(N)\)-optimal
277440.gz3 277440gz2 \([0, 1, 0, -1954398721, 32909749248479]\) \(521902963282042184836/6241849278890625\) \(9873854801957532754944000000\) \([2, 2]\) \(247726080\) \(4.1830\)  
277440.gz1 277440gz3 \([0, 1, 0, -31180067041, 2119149011469695]\) \(1059623036730633329075378/154307373046875\) \(488191408750944000000000000\) \([2]\) \(495452160\) \(4.5295\)  
277440.gz2 277440gz4 \([0, 1, 0, -3624818721, -31681049059521]\) \(1664865424893526702418/826424127435466125\) \(2614607137896969881432899584000\) \([2]\) \(495452160\) \(4.5295\)