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Properties

Label 48450b
Number of curves $4$
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, -17988025, 29357033125]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -17988025, 29357033125]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -17988025, 29357033125]) 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 48450b 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
\(11\) \( 1 + 5 T + 11 T^{2}\) 1.11.f
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 - 5 T + 29 T^{2}\) 1.29.af
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 48450.2.a.b

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);
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} - 4 q^{7} - q^{8} + q^{9} + 4 q^{11} - q^{12} + 2 q^{13} + 4 q^{14} + q^{16} - q^{17} - q^{18} - q^{19} + O(q^{20})\) 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 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 48450b

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.a3 48450b1 \([1, 1, 0, -17988025, 29357033125]\) \(41195916697879355491729/36197498880000\) \(565585920000000000\) \([2]\) \(2949120\) \(2.7064\) \(\Gamma_0(N)\)-optimal
48450.a2 48450b2 \([1, 1, 0, -18116025, 28917865125]\) \(42081620701292477662609/1220273715600000000\) \(19066776806250000000000\) \([2, 2]\) \(5898240\) \(3.0530\)  
48450.a4 48450b3 \([1, 1, 0, 4383975, 96035365125]\) \(596358945261507937391/255327150374524980000\) \(-3989486724601952812500000\) \([2]\) \(11796480\) \(3.3996\)  
48450.a1 48450b4 \([1, 1, 0, -42664025, -66303826875]\) \(549653727492794875187089/196605747070312500000\) \(3071964797973632812500000\) \([2]\) \(11796480\) \(3.3996\)