Properties

Label 494190co
Number of curves $4$
Conductor $494190$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("co1")
 
E.isogeny_class()
 

Elliptic curves in class 494190co

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
494190.co4 494190co1 \([1, -1, 0, 24280281, 97911713533]\) \(89962967236397039/287450726400000\) \(-5058065710340908646400000\) \([2]\) \(98304000\) \(3.4235\) \(\Gamma_0(N)\)-optimal*
494190.co3 494190co2 \([1, -1, 0, -228744999, 1147308759805]\) \(75224183150104868881/11219310000000000\) \(197418207688637310000000000\) \([2]\) \(196608000\) \(3.7700\) \(\Gamma_0(N)\)-optimal*
494190.co2 494190co3 \([1, -1, 0, -8587110519, 306282282600973]\) \(-3979640234041473454886161/1471455901872240\) \(-25892161535823949772544240\) \([2]\) \(491520000\) \(4.2282\) \(\Gamma_0(N)\)-optimal*
494190.co1 494190co4 \([1, -1, 0, -137393780499, 19601959388282905]\) \(16300610738133468173382620881/2228489100\) \(39213135564991469100\) \([2]\) \(983040000\) \(4.5747\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 494190co1.

Rank

sage: E.rank()
 

The elliptic curves in class 494190co have rank \(1\).

Complex multiplication

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

Modular form 494190.2.a.co

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} + 2 q^{7} - q^{8} - q^{10} + 2 q^{11} + 4 q^{13} - 2 q^{14} + q^{16} - q^{19} + O(q^{20})\) Copy content 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.