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Properties

Label 490110dd
Number of curves $8$
Conductor $490110$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("dd1")
 
E.isogeny_class()
 

Elliptic curves in class 490110dd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
490110.dd6 490110dd1 \([1, 0, 0, -76900, -10091248]\) \(-56667352321/16711680\) \(-14831677515694080\) \([2]\) \(3686400\) \(1.8174\) \(\Gamma_0(N)\)-optimal*
490110.dd5 490110dd2 \([1, 0, 0, -1306980, -575190000]\) \(278202094583041/16646400\) \(14773741275398400\) \([2, 2]\) \(7372800\) \(2.1640\) \(\Gamma_0(N)\)-optimal*
490110.dd4 490110dd3 \([1, 0, 0, -1383860, -503737728]\) \(330240275458561/67652010000\) \(60041407902048810000\) \([2, 2]\) \(14745600\) \(2.5106\) \(\Gamma_0(N)\)-optimal*
490110.dd2 490110dd4 \([1, 0, 0, -20911380, -36808042080]\) \(1139466686381936641/4080\) \(3621015018480\) \([2]\) \(14745600\) \(2.5106\)  
490110.dd3 490110dd5 \([1, 0, 0, -6938440, 6587239100]\) \(41623544884956481/2962701562500\) \(2629408542423201562500\) \([2, 2]\) \(29491200\) \(2.8572\) \(\Gamma_0(N)\)-optimal*
490110.dd7 490110dd6 \([1, 0, 0, 2940640, -3021461628]\) \(3168685387909439/6278181696900\) \(-5571909365985576288900\) \([2]\) \(29491200\) \(2.8572\)  
490110.dd1 490110dd7 \([1, 0, 0, -109044690, 438272042850]\) \(161572377633716256481/914742821250\) \(811837621027700021250\) \([2]\) \(58982400\) \(3.2037\) \(\Gamma_0(N)\)-optimal*
490110.dd8 490110dd8 \([1, 0, 0, 6294530, 28747170662]\) \(31077313442863199/420227050781250\) \(-372953054424133300781250\) \([2]\) \(58982400\) \(3.2037\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 5 curves highlighted, and conditionally curve 490110dd1.

Rank

sage: E.rank()
 

The elliptic curves in class 490110dd have rank \(1\).

Complex multiplication

The elliptic curves in class 490110dd do not have complex multiplication.

Modular form 490110.2.a.dd

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

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.