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Properties

Label 48450.c
Number of curves $2$
Conductor $48450$
CM no
Rank $1$
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, 0, -17323750, 27747428500]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -17323750, 27747428500]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -17323750, 27747428500]) 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.c have rank \(1\).

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 \(\mathbb{F}_p\)
\(7\) \( 1 + 3 T + 7 T^{2}\) 1.7.d
\(11\) \( 1 - 2 T + 11 T^{2}\) 1.11.ac
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(23\) \( 1 - 6 T + 23 T^{2}\) 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 48450.c do not have complex multiplication.

Modular form 48450.2.a.c

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

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

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.c

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.c1 48450a1 \([1, 1, 0, -17323750, 27747428500]\) \(-36798443442923099464801/2423324873327616\) \(-37864451145744000000\) \([]\) \(3192000\) \(2.8130\) \(\Gamma_0(N)\)-optimal
48450.c2 48450a2 \([1, 1, 0, 89954750, 64662121000]\) \(5152001506110026101064159/3096949914094458852996\) \(-48389842407725919578062500\) \([]\) \(15960000\) \(3.6177\)