Properties

Label 494190v
Number of curves $4$
Conductor $494190$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("v1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 494190v

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
494190.v3 494190v1 \([1, -1, 0, -80685, -8748459]\) \(3301293169/22800\) \(401195361862800\) \([2]\) \(2621440\) \(1.6357\) \(\Gamma_0(N)\)-optimal*
494190.v2 494190v2 \([1, -1, 0, -132705, 3954825]\) \(14688124849/8122500\) \(142925847663622500\) \([2, 2]\) \(5242880\) \(1.9823\) \(\Gamma_0(N)\)-optimal*
494190.v1 494190v3 \([1, -1, 0, -1615275, 789420411]\) \(26487576322129/44531250\) \(783584691138281250\) \([2]\) \(10485760\) \(2.3288\) \(\Gamma_0(N)\)-optimal*
494190.v4 494190v4 \([1, -1, 0, 517545, 30875175]\) \(871257511151/527800050\) \(-9287321581182190050\) \([2]\) \(10485760\) \(2.3288\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 494190v1.

Rank

sage: E.rank()
 

The elliptic curves in class 494190v have rank \(0\).

Complex multiplication

The elliptic curves in class 494190v do not have complex multiplication.

Modular form 494190.2.a.v

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{5} - q^{8} + q^{10} + 4q^{11} + 2q^{13} + q^{16} - q^{19} + O(q^{20})\)  Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) 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

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.