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Properties

Label 48450.x
Number of curves $4$
Conductor $48450$
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, -365923063, 2694065050781]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -365923063, 2694065050781]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -365923063, 2694065050781]) 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 48450.x have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 48450.x do not have complex multiplication.

Modular form 48450.2.a.x

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

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

Isogeny graph

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

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

Elliptic curves in class 48450.x

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
48450.x1 48450ba4 \([1, 1, 1, -365923063, 2694065050781]\) \(346795165011870675497264041/121778756846679600\) \(1902793075729368750000\) \([2]\) \(17694720\) \(3.4381\)  
48450.x2 48450ba2 \([1, 1, 1, -22973063, 41689750781]\) \(85814444987865209552041/1585867720793760000\) \(24779183137402500000000\) \([2, 2]\) \(8847360\) \(3.0915\)  
48450.x3 48450ba1 \([1, 1, 1, -2973063, -990249219]\) \(186001322269702352041/80595993600000000\) \(1259312400000000000000\) \([2]\) \(4423680\) \(2.7449\) \(\Gamma_0(N)\)-optimal
48450.x4 48450ba3 \([1, 1, 1, -23063, 121234450781]\) \(-86826493040041/406364619140547159600\) \(-6349447174071049368750000\) \([2]\) \(17694720\) \(3.4381\)