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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 0, -3177978, 2179257003]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -3177978, 2179257003]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -3177978, 2179257003]) 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 150765.s have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1 + T\)
\(5\)\(1 + T\)
\(19\)\(1 - T\)
\(23\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 150765.s do not have complex multiplication.

Modular form 150765.2.a.s

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);
 
\(q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} - 4 q^{7} - 3 q^{8} + q^{9} - q^{10} - 4 q^{11} + q^{12} + 2 q^{13} - 4 q^{14} + q^{15} - q^{16} - 2 q^{17} + q^{18} + q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

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

Elliptic curves in class 150765.s

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
150765.s1 150765w3 \([1, 1, 0, -3177978, 2179257003]\) \(23977812996389881/146611125\) \(21703708226665125\) \([2]\) \(3548160\) \(2.3219\)  
150765.s2 150765w4 \([1, 1, 0, -654648, -165373623]\) \(209595169258201/41748046875\) \(6180209233154296875\) \([2]\) \(3548160\) \(2.3219\)  
150765.s3 150765w2 \([1, 1, 0, -202353, 32641128]\) \(6189976379881/456890625\) \(67636209847640625\) \([2, 2]\) \(1774080\) \(1.9753\)  
150765.s4 150765w1 \([1, 1, 0, 11892, 2261187]\) \(1256216039/15582375\) \(-2306750735856375\) \([2]\) \(887040\) \(1.6287\) \(\Gamma_0(N)\)-optimal