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Properties

Label 363a
Number of curves $4$
Conductor $363$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 1, -789, 8130]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -789, 8130]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -789, 8130]) 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 363a have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1 + T\)
\(11\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + 2 T + 2 T^{2}\) 1.2.c
\(5\) \( 1 - 4 T + 5 T^{2}\) 1.5.ae
\(7\) \( 1 - T + 7 T^{2}\) 1.7.ab
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 4 T + 17 T^{2}\) 1.17.ae
\(19\) \( 1 + 3 T + 19 T^{2}\) 1.19.d
\(23\) \( 1 - 2 T + 23 T^{2}\) 1.23.ac
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 363a do not have complex multiplication.

Modular form 363.2.a.a

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

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

Isogeny graph

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

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

Elliptic curves in class 363a

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
363.b3 363a1 \([1, 1, 1, -789, 8130]\) \(30664297/297\) \(526153617\) \([4]\) \(180\) \(0.49317\) \(\Gamma_0(N)\)-optimal
363.b2 363a2 \([1, 1, 1, -1394, -6874]\) \(169112377/88209\) \(156267624249\) \([2, 2]\) \(360\) \(0.83974\)  
363.b1 363a3 \([1, 1, 1, -17729, -915100]\) \(347873904937/395307\) \(700310464227\) \([2]\) \(720\) \(1.1863\)  
363.b4 363a4 \([1, 1, 1, 5261, -46804]\) \(9090072503/5845851\) \(-10356281643411\) \([2]\) \(720\) \(1.1863\)