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Properties

Label 54450dh
Number of curves $4$
Conductor $54450$
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, 0, -82242, -12327584]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -82242, -12327584]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -82242, -12327584]) 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 54450dh have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 54450dh do not have complex multiplication.

Modular form 54450.2.a.dh

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);
 
\(q - q^{2} + q^{4} - 2 q^{7} - q^{8} - 6 q^{13} + 2 q^{14} + q^{16} + 2 q^{17} + O(q^{20})\) 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 54450dh

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
54450.v4 54450dh1 \([1, -1, 0, -82242, -12327584]\) \(-24389/12\) \(-30268780523437500\) \([2]\) \(448000\) \(1.8677\) \(\Gamma_0(N)\)-optimal
54450.v2 54450dh2 \([1, -1, 0, -1443492, -667088834]\) \(131872229/18\) \(45403170785156250\) \([2]\) \(896000\) \(2.2143\)  
54450.v3 54450dh3 \([1, -1, 0, -762867, 1232535541]\) \(-19465109/248832\) \(-627653432934000000000\) \([2]\) \(2240000\) \(2.6724\)  
54450.v1 54450dh4 \([1, -1, 0, -22542867, 41068155541]\) \(502270291349/1889568\) \(4766243256342562500000\) \([2]\) \(4480000\) \(3.0190\)