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Properties

Label 494214v
Number of curves $2$
Conductor $494214$
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, -5479254, -4910983020]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -5479254, -4910983020]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -5479254, -4910983020]) 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 494214v have rank \(0\).

Complex multiplication

The elliptic curves in class 494214v do not have complex multiplication.

Modular form 494214.2.a.v

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);
 
\(q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} - q^{8} + q^{9} - 2 q^{10} + 4 q^{11} - q^{12} - 4 q^{13} - 2 q^{15} + q^{16} - 2 q^{17} - q^{18} - 8 q^{19} + O(q^{20})\) 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.

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

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 494214v

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
494214.v1 494214v1 \([1, 1, 0, -5479254, -4910983020]\) \(32553430057/212544\) \(118779154623608786496\) \([2]\) \(29030400\) \(2.6879\) \(\Gamma_0(N)\)-optimal
494214.v2 494214v2 \([1, 1, 0, -2184494, -10756546212]\) \(-2062933417/88232328\) \(-49308196563125597494152\) \([2]\) \(58060800\) \(3.0345\)