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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 0, -40671, 3167194]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -40671, 3167194]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -40671, 3167194]) 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 693.d have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1\)
\(7\)\(1 - T\)
\(11\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 693.d do not have complex multiplication.

Modular form 693.2.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);
 
\(q + q^{2} - q^{4} + 2 q^{5} + q^{7} - 3 q^{8} + 2 q^{10} + q^{11} + 6 q^{13} + q^{14} - q^{16} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 693.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
693.d1 693d5 \([1, -1, 0, -40671, 3167194]\) \(10206027697760497/5557167\) \(4051174743\) \([2]\) \(1280\) \(1.1721\)  
693.d2 693d3 \([1, -1, 0, -2556, 49387]\) \(2533811507137/58110129\) \(42362284041\) \([2, 2]\) \(640\) \(0.82552\)  
693.d3 693d2 \([1, -1, 0, -351, -1328]\) \(6570725617/2614689\) \(1906108281\) \([2, 2]\) \(320\) \(0.47895\)  
693.d4 693d1 \([1, -1, 0, -306, -1985]\) \(4354703137/1617\) \(1178793\) \([2]\) \(160\) \(0.13238\) \(\Gamma_0(N)\)-optimal
693.d5 693d6 \([1, -1, 0, 279, 150880]\) \(3288008303/13504609503\) \(-9844860327687\) \([2]\) \(1280\) \(1.1721\)  
693.d6 693d4 \([1, -1, 0, 1134, -10535]\) \(221115865823/190238433\) \(-138683817657\) \([2]\) \(640\) \(0.82552\)