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Properties

Label 219450d
Number of curves $4$
Conductor $219450$
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, 0, 0, -14102053, -18455059903]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -14102053, -18455059903]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -14102053, -18455059903]) 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 219450d have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(5\)\(1\)
\(7\)\(1 + T\)
\(11\)\(1 + T\)
\(19\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 219450d do not have complex multiplication.

Modular form 219450.2.a.d

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

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

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 219450d

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
219450.hl3 219450d1 \([1, 0, 0, -14102053, -18455059903]\) \(2481194036785531116697829/258949680664193563968\) \(32368710083024195496000\) \([2]\) \(27264000\) \(3.0543\) \(\Gamma_0(N)\)-optimal
219450.hl4 219450d2 \([1, 0, 0, 18108147, -90509277303]\) \(5253342688178294786187931/31352380501055976236952\) \(-3919047562631997029619000\) \([2]\) \(54528000\) \(3.4009\)  
219450.hl1 219450d3 \([1, 0, 0, -1963596578, 33490702261572]\) \(6698391064416261144129516088949/2449861814311786119168\) \(306232726788973264896000\) \([10]\) \(136320000\) \(3.8590\)  
219450.hl2 219450d4 \([1, 0, 0, -1954585378, 33813312232772]\) \(-6606594261153843534370179395189/128160831539202997006467072\) \(-16020103942400374625808384000\) \([10]\) \(272640000\) \(4.2056\)