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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, -3864, -204480]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -3864, -204480]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -3864, -204480]) 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 40362bc have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(7\)\(1 + T\)
\(31\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - T + 5 T^{2}\) 1.5.ab
\(11\) \( 1 - 5 T + 11 T^{2}\) 1.11.af
\(13\) \( 1 - 5 T + 13 T^{2}\) 1.13.af
\(17\) \( 1 - 3 T + 17 T^{2}\) 1.17.ad
\(19\) \( 1 + T + 19 T^{2}\) 1.19.b
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 40362bc do not have complex multiplication.

Modular form 40362.2.a.bc

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

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

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 40362bc

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
40362.bb5 40362bc1 \([1, 0, 0, -3864, -204480]\) \(-7189057/16128\) \(-14313659367168\) \([2]\) \(122880\) \(1.2129\) \(\Gamma_0(N)\)-optimal
40362.bb4 40362bc2 \([1, 0, 0, -80744, -8830416]\) \(65597103937/63504\) \(56360033758224\) \([2, 2]\) \(245760\) \(1.5595\)  
40362.bb3 40362bc3 \([1, 0, 0, -99964, -4313716]\) \(124475734657/63011844\) \(55923243496597764\) \([2, 2]\) \(491520\) \(1.9061\)  
40362.bb1 40362bc4 \([1, 0, 0, -1291604, -565099500]\) \(268498407453697/252\) \(223650927612\) \([2]\) \(491520\) \(1.9061\)  
40362.bb6 40362bc5 \([1, 0, 0, 370926, -33226362]\) \(6359387729183/4218578658\) \(-3744004087563040098\) \([2]\) \(983040\) \(2.2527\)  
40362.bb2 40362bc6 \([1, 0, 0, -878374, 313744610]\) \(84448510979617/933897762\) \(828837701452661922\) \([2]\) \(983040\) \(2.2527\)